Web_scraping_with_beautiful_soup_and_pandas_complete.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Web Scraping with Beautiful Soup and Pandas\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Web scraping is the process of using bots to extract content and data from a website.\n",
    "\n",
    "Unlike screen scraping, which only copies pixels displayed onscreen, web scraping extracts underlying HTML code and, with it, data stored in a database. The scraper can then replicate entire website content elsewhere.\n",
    "\n",
    "Web scraping is used in a variety of digital businesses that rely on data harvesting. Legitimate use cases include:\n",
    "\n",
    "- Search engine bots crawling a site, analyzing its content and then ranking it.\n",
    "- Price comparison sites deploying bots to auto-fetch prices and product descriptions for allied seller websites.\n",
    "- Market research companies using scrapers to pull data from forums and social media (e.g., for sentiment analysis)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Table of Contents\n",
    "[1. Making Database From Scratch With Beautiful Soup](#I.-Making-Database-From-Scratch-With-Beautiful-Soup) <br>\n",
    "- [Scrape The Data](#Scrape-The-Data) \n",
    "- [Make A Database](#Make-A-Database)\n",
    "\n",
    "[2. Web Scraping Using Pandas](#II.-Web-Scraping-Using-Pandas) <br>\n",
    "- [Get The URL](#Get-The-URL) \n",
    "- [Read The HTML Webpage Into Pandas](#Read-The-HTML-Webpage-Into-Pandas)\n",
    "- [Data Cleaning](#Data-Cleaning) \n",
    "- [Quick Exploratory Data Analysis](#Quick-Exploratory-Data-Analysis)<br>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## I. Making Database From Scratch With Beautiful Soup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are a number of different packages available for web scraping, and one of the most popular is [Beautiful Soup](https://www.crummy.com/software/BeautifulSoup/bs4/doc/). Beautiful Soup parses web content into a Python object and makes the [DOM](https://en.wikipedia.org/wiki/Document_Object_Model) queryable element by element. Used in conjunction with a requests package, it makes web scraping very easy!"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "### Installation of Beautiful Soup (if haven't done so)\n",
    "In the `bash` terminal or `Anaconda Prompt`,run:\n",
    "```bash\n",
    "conda install beautifulsoup4\n",
    "```\n",
    "---"
   ]
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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Standard imports\n",
    "import pandas as pd\n",
    "\n",
    "# For web scraping\n",
    "import requests\n",
    "import urllib.request\n",
    "from bs4 import BeautifulSoup\n",
    "\n",
    "# For performing regex operations\n",
    "import re\n",
    "\n",
    "# Data visualization\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this kickoff, we'll be scraping the **random functions names and usages** from Python documentation from the website [docs.python.org](https://docs.python.org/)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scrape The Data\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save the URL of the webpage we want to scrape to a variable\n",
    "url = 'https://docs.python.org/3/library/random.html#module-random'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When web scraping, the first step is to pull down the content of the page into a Python (string) variable. For simpler webscraping tasks you can do this with the `requests` package, which is what we'll use here. For more complex tasks (involving, e.g., webpages with lots of Javascript or other elements that are rendered by the web browser) you may need to use something more advanced, like `urllib` or [Selenium](https://selenium-python.readthedocs.io/index.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Send a get request and assign the response to a variable\n",
    "response = requests.get(url)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at what we have!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Response [200]>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "response"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
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    "tags": []
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   "outputs": [
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       "b'\\n<!DOCTYPE html>\\n\\n<html xmlns=\"http://www.w3.org/1999/xhtml\">\\n  <head>\\n    <meta charset=\"utf-8\" />\\n    <title>random \\xe2\\x80\\x94 Generate pseudo-random numbers &#8212; Python 3.9.2 documentation</title>\\n    <link rel=\"stylesheet\" href=\"../_static/pydoctheme.css\" type=\"text/css\" />\\n    <link rel=\"stylesheet\" href=\"../_static/pygments.css\" type=\"text/css\" />\\n    \\n    <script id=\"documentation_options\" data-url_root=\"../\" src=\"../_static/documentation_options.js\"></script>\\n    <script src=\"../_static/jquery.js\"></script>\\n    <script src=\"../_static/underscore.js\"></script>\\n    <script src=\"../_static/doctools.js\"></script>\\n    <script src=\"../_static/language_data.js\"></script>\\n    \\n    <script src=\"../_static/sidebar.js\"></script>\\n    \\n    <link rel=\"search\" type=\"application/opensearchdescription+xml\"\\n          title=\"Search within Python 3.9.2 documentation\"\\n          href=\"../_static/opensearch.xml\"/>\\n    <link rel=\"author\" title=\"About these documents\" href=\"../about.html\" />\\n    <link rel=\"index\" title=\"Index\" href=\"../genindex.html\" />\\n    <link rel=\"search\" title=\"Search\" href=\"../search.html\" />\\n    <link rel=\"copyright\" title=\"Copyright\" href=\"../copyright.html\" />\\n    <link rel=\"next\" title=\"statistics \\xe2\\x80\\x94 Mathematical statistics functions\" href=\"statistics.html\" />\\n    <link rel=\"prev\" title=\"fractions \\xe2\\x80\\x94 Rational numbers\" href=\"fractions.html\" />\\n    <link rel=\"canonical\" href=\"https://docs.python.org/3/library/random.html\" />\\n    \\n      \\n      \\n    \\n\\n    \\n    <style>\\n      @media only screen {\\n        table.full-width-table {\\n            width: 100%;\\n        }\\n      }\\n    </style>\\n\\n    <link rel=\"shortcut icon\" type=\"image/png\" href=\"../_static/py.png\" />\\n    \\n    <script type=\"text/javascript\" src=\"../_static/copybutton.js\"></script>\\n    \\n     \\n\\n\\n  </head><body>\\n  \\n    <div class=\"related\" role=\"navigation\" aria-label=\"related navigation\">\\n      <h3>Navigation</h3>\\n      <ul>\\n        <li class=\"right\" style=\"margin-right: 10px\">\\n          <a href=\"../genindex.html\" title=\"General Index\"\\n             accesskey=\"I\">index</a></li>\\n        <li class=\"right\" >\\n          <a href=\"../py-modindex.html\" title=\"Python Module Index\"\\n             >modules</a> |</li>\\n        <li class=\"right\" >\\n          <a href=\"statistics.html\" title=\"statistics \\xe2\\x80\\x94 Mathematical statistics functions\"\\n             accesskey=\"N\">next</a> |</li>\\n        <li class=\"right\" >\\n          <a href=\"fractions.html\" title=\"fractions \\xe2\\x80\\x94 Rational numbers\"\\n             accesskey=\"P\">previous</a> |</li>\\n\\n    <li><img src=\"../_static/py.png\" alt=\"\"\\n             style=\"vertical-align: middle; margin-top: -1px\"/></li>\\n    <li><a href=\"https://www.python.org/\">Python</a> &#187;</li>\\n    \\n\\n    <li>\\n      <a href=\"../index.html\">3.9.2 Documentation</a> &#187;\\n    </li>\\n\\n          <li class=\"nav-item nav-item-1\"><a href=\"index.html\" >The Python Standard Library</a> &#187;</li>\\n          <li class=\"nav-item nav-item-2\"><a href=\"numeric.html\" accesskey=\"U\">Numeric and Mathematical Modules</a> &#187;</li>\\n    <li class=\"right\">\\n        \\n\\n    <div class=\"inline-search\" style=\"display: none\" role=\"search\">\\n        <form class=\"inline-search\" action=\"../search.html\" method=\"get\">\\n          <input placeholder=\"Quick search\" type=\"text\" name=\"q\" />\\n          <input type=\"submit\" value=\"Go\" />\\n          <input type=\"hidden\" name=\"check_keywords\" value=\"yes\" />\\n          <input type=\"hidden\" name=\"area\" value=\"default\" />\\n        </form>\\n    </div>\\n    <script type=\"text/javascript\">$(\\'.inline-search\\').show(0);</script>\\n         |\\n    </li>\\n\\n      </ul>\\n    </div>    \\n\\n    <div class=\"document\">\\n      <div class=\"documentwrapper\">\\n        <div class=\"bodywrapper\">\\n          <div class=\"body\" role=\"main\">\\n            \\n  <div class=\"section\" id=\"module-random\">\\n<span id=\"random-generate-pseudo-random-numbers\"></span><h1><a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code></a> \\xe2\\x80\\x94 Generate pseudo-random numbers<a class=\"headerlink\" href=\"#module-random\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h1>\\n<p><strong>Source code:</strong> <a class=\"reference external\" href=\"https://github.com/python/cpython/tree/3.9/Lib/random.py\">Lib/random.py</a></p>\\n<hr class=\"docutils\" />\\n<p>This module implements pseudo-random number generators for various\\ndistributions.</p>\\n<p>For integers, there is uniform selection from a range. For sequences, there is\\nuniform selection of a random element, a function to generate a random\\npermutation of a list in-place, and a function for random sampling without\\nreplacement.</p>\\n<p>On the real line, there are functions to compute uniform, normal (Gaussian),\\nlognormal, negative exponential, gamma, and beta distributions. For generating\\ndistributions of angles, the von Mises distribution is available.</p>\\n<p>Almost all module functions depend on the basic function <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a>, which\\ngenerates a random float uniformly in the semi-open range [0.0, 1.0).  Python\\nuses the Mersenne Twister as the core generator.  It produces 53-bit precision\\nfloats and has a period of 2**19937-1.  The underlying implementation in C is\\nboth fast and threadsafe.  The Mersenne Twister is one of the most extensively\\ntested random number generators in existence.  However, being completely\\ndeterministic, it is not suitable for all purposes, and is completely unsuitable\\nfor cryptographic purposes.</p>\\n<p>The functions supplied by this module are actually bound methods of a hidden\\ninstance of the <a class=\"reference internal\" href=\"#random.Random\" title=\"random.Random\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">random.Random</span></code></a> class.  You can instantiate your own\\ninstances of <a class=\"reference internal\" href=\"#random.Random\" title=\"random.Random\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">Random</span></code></a> to get generators that don\\xe2\\x80\\x99t share state.</p>\\n<p>Class <a class=\"reference internal\" href=\"#random.Random\" title=\"random.Random\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">Random</span></code></a> can also be subclassed if you want to use a different\\nbasic generator of your own devising: in that case, override the <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">random()</span></code>,\\n<code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">seed()</span></code>, <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getstate()</span></code>, and <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">setstate()</span></code> methods.\\nOptionally, a new generator can supply a <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getrandbits()</span></code> method \\xe2\\x80\\x94 this\\nallows <a class=\"reference internal\" href=\"#random.randrange\" title=\"random.randrange\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">randrange()</span></code></a> to produce selections over an arbitrarily large range.</p>\\n<p>The <a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code></a> module also provides the <a class=\"reference internal\" href=\"#random.SystemRandom\" title=\"random.SystemRandom\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">SystemRandom</span></code></a> class which\\nuses the system function <a class=\"reference internal\" href=\"os.html#os.urandom\" title=\"os.urandom\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">os.urandom()</span></code></a> to generate random numbers\\nfrom sources provided by the operating system.</p>\\n<div class=\"admonition warning\">\\n<p class=\"admonition-title\">Warning</p>\\n<p>The pseudo-random generators of this module should not be used for\\nsecurity purposes.  For security or cryptographic uses, see the\\n<a class=\"reference internal\" href=\"secrets.html#module-secrets\" title=\"secrets: Generate secure random numbers for managing secrets.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">secrets</span></code></a> module.</p>\\n</div>\\n<div class=\"admonition seealso\">\\n<p class=\"admonition-title\">See also</p>\\n<p>M. Matsumoto and T. Nishimura, \\xe2\\x80\\x9cMersenne Twister: A 623-dimensionally\\nequidistributed uniform pseudorandom number generator\\xe2\\x80\\x9d, ACM Transactions on\\nModeling and Computer Simulation Vol. 8, No. 1, January pp.3\\xe2\\x80\\x9330 1998.</p>\\n<p><a class=\"reference external\" href=\"https://code.activestate.com/recipes/576707/\">Complementary-Multiply-with-Carry recipe</a> for a compatible alternative\\nrandom number generator with a long period and comparatively simple update\\noperations.</p>\\n</div>\\n<div class=\"section\" id=\"bookkeeping-functions\">\\n<h2>Bookkeeping functions<a class=\"headerlink\" href=\"#bookkeeping-functions\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<dl class=\"function\">\\n<dt id=\"random.seed\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">seed</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a=None</em>, <em class=\"sig-param\">version=2</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.seed\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Initialize the random number generator.</p>\\n<p>If <em>a</em> is omitted or <code class=\"docutils literal notranslate\"><span class=\"pre\">None</span></code>, the current system time is used.  If\\nrandomness sources are provided by the operating system, they are used\\ninstead of the system time (see the <a class=\"reference internal\" href=\"os.html#os.urandom\" title=\"os.urandom\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">os.urandom()</span></code></a> function for details\\non availability).</p>\\n<p>If <em>a</em> is an int, it is used directly.</p>\\n<p>With version 2 (the default), a <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a>, <a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a>, or <a class=\"reference internal\" href=\"stdtypes.html#bytearray\" title=\"bytearray\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytearray</span></code></a>\\nobject gets converted to an <a class=\"reference internal\" href=\"functions.html#int\" title=\"int\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">int</span></code></a> and all of its bits are used.</p>\\n<p>With version 1 (provided for reproducing random sequences from older versions\\nof Python), the algorithm for <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a> and <a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a> generates a\\nnarrower range of seeds.</p>\\n<div class=\"versionchanged\">\\n<p><span class=\"versionmodified changed\">Changed in version 3.2: </span>Moved to the version 2 scheme which uses all of the bits in a string seed.</p>\\n</div>\\n<div class=\"deprecated\">\\n<p><span class=\"versionmodified deprecated\">Deprecated since version 3.9: </span>In the future, the <em>seed</em> must be one of the following types:\\n<em>NoneType</em>, <a class=\"reference internal\" href=\"functions.html#int\" title=\"int\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">int</span></code></a>, <a class=\"reference internal\" href=\"functions.html#float\" title=\"float\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">float</span></code></a>, <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a>,\\n<a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a>, or <a class=\"reference internal\" href=\"stdtypes.html#bytearray\" title=\"bytearray\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytearray</span></code></a>.</p>\\n</div>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.getstate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">getstate</code><span class=\"sig-paren\">(</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.getstate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return an object capturing the current internal state of the generator.  This\\nobject can be passed to <a class=\"reference internal\" href=\"#random.setstate\" title=\"random.setstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">setstate()</span></code></a> to restore the state.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.setstate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">setstate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">state</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.setstate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p><em>state</em> should have been obtained from a previous call to <a class=\"reference internal\" href=\"#random.getstate\" title=\"random.getstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">getstate()</span></code></a>, and\\n<a class=\"reference internal\" href=\"#random.setstate\" title=\"random.setstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">setstate()</span></code></a> restores the internal state of the generator to what it was at\\nthe time <a class=\"reference internal\" href=\"#random.getstate\" title=\"random.getstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">getstate()</span></code></a> was called.</p>\\n</dd></dl>\\n\\n</div>\\n<div class=\"section\" id=\"functions-for-bytes\">\\n<h2>Functions for bytes<a class=\"headerlink\" href=\"#functions-for-bytes\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<dl class=\"function\">\\n<dt id=\"random.randbytes\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randbytes</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">n</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randbytes\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Generate <em>n</em> random bytes.</p>\\n<p>This method should not be used for generating security tokens.\\nUse <a class=\"reference internal\" href=\"secrets.html#secrets.token_bytes\" title=\"secrets.token_bytes\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">secrets.token_bytes()</span></code></a> instead.</p>\\n<div class=\"versionadded\">\\n<p><span class=\"versionmodified added\">New in version 3.9.</span></p>\\n</div>\\n</dd></dl>\\n\\n</div>\\n<div class=\"section\" id=\"functions-for-integers\">\\n<h2>Functions for integers<a class=\"headerlink\" href=\"#functions-for-integers\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<dl class=\"function\">\\n<dt id=\"random.randrange\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randrange</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">stop</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randrange\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dt>\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randrange</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">start</em>, <em class=\"sig-param\">stop</em><span class=\"optional\">[</span>, <em class=\"sig-param\">step</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span></dt>\\n<dd><p>Return a randomly selected element from <code class=\"docutils literal notranslate\"><span class=\"pre\">range(start,</span> <span class=\"pre\">stop,</span> <span class=\"pre\">step)</span></code>.  This is\\nequivalent to <code class=\"docutils literal notranslate\"><span class=\"pre\">choice(range(start,</span> <span class=\"pre\">stop,</span> <span class=\"pre\">step))</span></code>, but doesn\\xe2\\x80\\x99t actually build a\\nrange object.</p>\\n<p>The positional argument pattern matches that of <a class=\"reference internal\" href=\"stdtypes.html#range\" title=\"range\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">range()</span></code></a>.  Keyword arguments\\nshould not be used because the function may use them in unexpected ways.</p>\\n<div class=\"versionchanged\">\\n<p><span class=\"versionmodified changed\">Changed in version 3.2: </span><a class=\"reference internal\" href=\"#random.randrange\" title=\"random.randrange\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">randrange()</span></code></a> is more sophisticated about producing equally distributed\\nvalues.  Formerly it used a style like <code class=\"docutils literal notranslate\"><span class=\"pre\">int(random()*n)</span></code> which could produce\\nslightly uneven distributions.</p>\\n</div>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.randint\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randint</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a</em>, <em class=\"sig-param\">b</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randint\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return a random integer <em>N</em> such that <code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">b</span></code>.  Alias for\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">randrange(a,</span> <span class=\"pre\">b+1)</span></code>.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.getrandbits\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">getrandbits</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">k</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.getrandbits\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Returns a non-negative Python integer with <em>k</em> random bits. This method\\nis supplied with the MersenneTwister generator and some other generators\\nmay also provide it as an optional part of the API. When available,\\n<a class=\"reference internal\" href=\"#random.getrandbits\" title=\"random.getrandbits\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getrandbits()</span></code></a> enables <a class=\"reference internal\" href=\"#random.randrange\" title=\"random.randrange\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">randrange()</span></code></a> to handle arbitrarily large\\nranges.</p>\\n<div class=\"versionchanged\">\\n<p><span class=\"versionmodified changed\">Changed in version 3.9: </span>This method now accepts zero for <em>k</em>.</p>\\n</div>\\n</dd></dl>\\n\\n</div>\\n<div class=\"section\" id=\"functions-for-sequences\">\\n<h2>Functions for sequences<a class=\"headerlink\" href=\"#functions-for-sequences\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<dl class=\"function\">\\n<dt id=\"random.choice\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">choice</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">seq</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.choice\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return a random element from the non-empty sequence <em>seq</em>. If <em>seq</em> is empty,\\nraises <a class=\"reference internal\" href=\"exceptions.html#IndexError\" title=\"IndexError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">IndexError</span></code></a>.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.choices\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">choices</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">population</em>, <em class=\"sig-param\">weights=None</em>, <em class=\"sig-param\">*</em>, <em class=\"sig-param\">cum_weights=None</em>, <em class=\"sig-param\">k=1</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.choices\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return a <em>k</em> sized list of elements chosen from the <em>population</em> with replacement.\\nIf the <em>population</em> is empty, raises <a class=\"reference internal\" href=\"exceptions.html#IndexError\" title=\"IndexError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">IndexError</span></code></a>.</p>\\n<p>If a <em>weights</em> sequence is specified, selections are made according to the\\nrelative weights.  Alternatively, if a <em>cum_weights</em> sequence is given, the\\nselections are made according to the cumulative weights (perhaps computed\\nusing <a class=\"reference internal\" href=\"itertools.html#itertools.accumulate\" title=\"itertools.accumulate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">itertools.accumulate()</span></code></a>).  For example, the relative weights\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">[10,</span> <span class=\"pre\">5,</span> <span class=\"pre\">30,</span> <span class=\"pre\">5]</span></code> are equivalent to the cumulative weights\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">[10,</span> <span class=\"pre\">15,</span> <span class=\"pre\">45,</span> <span class=\"pre\">50]</span></code>.  Internally, the relative weights are converted to\\ncumulative weights before making selections, so supplying the cumulative\\nweights saves work.</p>\\n<p>If neither <em>weights</em> nor <em>cum_weights</em> are specified, selections are made\\nwith equal probability.  If a weights sequence is supplied, it must be\\nthe same length as the <em>population</em> sequence.  It is a <a class=\"reference internal\" href=\"exceptions.html#TypeError\" title=\"TypeError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">TypeError</span></code></a>\\nto specify both <em>weights</em> and <em>cum_weights</em>.</p>\\n<p>The <em>weights</em> or <em>cum_weights</em> can use any numeric type that interoperates\\nwith the <a class=\"reference internal\" href=\"functions.html#float\" title=\"float\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">float</span></code></a> values returned by <a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a> (that includes\\nintegers, floats, and fractions but excludes decimals).  Behavior is\\nundefined if any weight is negative.  A <a class=\"reference internal\" href=\"exceptions.html#ValueError\" title=\"ValueError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">ValueError</span></code></a> is raised if all\\nweights are zero.</p>\\n<p>For a given seed, the <a class=\"reference internal\" href=\"#random.choices\" title=\"random.choices\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choices()</span></code></a> function with equal weighting\\ntypically produces a different sequence than repeated calls to\\n<a class=\"reference internal\" href=\"#random.choice\" title=\"random.choice\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choice()</span></code></a>.  The algorithm used by <a class=\"reference internal\" href=\"#random.choices\" title=\"random.choices\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choices()</span></code></a> uses floating\\npoint arithmetic for internal consistency and speed.  The algorithm used\\nby <a class=\"reference internal\" href=\"#random.choice\" title=\"random.choice\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choice()</span></code></a> defaults to integer arithmetic with repeated selections\\nto avoid small biases from round-off error.</p>\\n<div class=\"versionadded\">\\n<p><span class=\"versionmodified added\">New in version 3.6.</span></p>\\n</div>\\n<div class=\"versionchanged\">\\n<p><span class=\"versionmodified changed\">Changed in version 3.9: </span>Raises a <a class=\"reference internal\" href=\"exceptions.html#ValueError\" title=\"ValueError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">ValueError</span></code></a> if all weights are zero.</p>\\n</div>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.shuffle\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">shuffle</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">x</em><span class=\"optional\">[</span>, <em class=\"sig-param\">random</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.shuffle\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Shuffle the sequence <em>x</em> in place.</p>\\n<p>The optional argument <em>random</em> is a 0-argument function returning a random\\nfloat in [0.0, 1.0); by default, this is the function <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a>.</p>\\n<p>To shuffle an immutable sequence and return a new shuffled list, use\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">sample(x,</span> <span class=\"pre\">k=len(x))</span></code> instead.</p>\\n<p>Note that even for small <code class=\"docutils literal notranslate\"><span class=\"pre\">len(x)</span></code>, the total number of permutations of <em>x</em>\\ncan quickly grow larger than the period of most random number generators.\\nThis implies that most permutations of a long sequence can never be\\ngenerated.  For example, a sequence of length 2080 is the largest that\\ncan fit within the period of the Mersenne Twister random number generator.</p>\\n<div class=\"deprecated-removed\">\\n<p><span class=\"versionmodified\">Deprecated since version 3.9, will be removed in version 3.11: </span>The optional parameter <em>random</em>.</p>\\n</div>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.sample\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">sample</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">population</em>, <em class=\"sig-param\">k</em>, <em class=\"sig-param\">*</em>, <em class=\"sig-param\">counts=None</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.sample\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return a <em>k</em> length list of unique elements chosen from the population sequence\\nor set. Used for random sampling without replacement.</p>\\n<p>Returns a new list containing elements from the population while leaving the\\noriginal population unchanged.  The resulting list is in selection order so that\\nall sub-slices will also be valid random samples.  This allows raffle winners\\n(the sample) to be partitioned into grand prize and second place winners (the\\nsubslices).</p>\\n<p>Members of the population need not be <a class=\"reference internal\" href=\"../glossary.html#term-hashable\"><span class=\"xref std std-term\">hashable</span></a> or unique.  If the population\\ncontains repeats, then each occurrence is a possible selection in the sample.</p>\\n<p>Repeated elements can be specified one at a time or with the optional\\nkeyword-only <em>counts</em> parameter.  For example, <code class=\"docutils literal notranslate\"><span class=\"pre\">sample([\\'red\\',</span> <span class=\"pre\">\\'blue\\'],</span>\\n<span class=\"pre\">counts=[4,</span> <span class=\"pre\">2],</span> <span class=\"pre\">k=5)</span></code> is equivalent to <code class=\"docutils literal notranslate\"><span class=\"pre\">sample([\\'red\\',</span> <span class=\"pre\">\\'red\\',</span> <span class=\"pre\">\\'red\\',</span> <span class=\"pre\">\\'red\\',</span>\\n<span class=\"pre\">\\'blue\\',</span> <span class=\"pre\">\\'blue\\'],</span> <span class=\"pre\">k=5)</span></code>.</p>\\n<p>To choose a sample from a range of integers, use a <a class=\"reference internal\" href=\"stdtypes.html#range\" title=\"range\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">range()</span></code></a> object as an\\nargument.  This is especially fast and space efficient for sampling from a large\\npopulation:  <code class=\"docutils literal notranslate\"><span class=\"pre\">sample(range(10000000),</span> <span class=\"pre\">k=60)</span></code>.</p>\\n<p>If the sample size is larger than the population size, a <a class=\"reference internal\" href=\"exceptions.html#ValueError\" title=\"ValueError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">ValueError</span></code></a>\\nis raised.</p>\\n<div class=\"versionchanged\">\\n<p><span class=\"versionmodified changed\">Changed in version 3.9: </span>Added the <em>counts</em> parameter.</p>\\n</div>\\n<div class=\"deprecated\">\\n<p><span class=\"versionmodified deprecated\">Deprecated since version 3.9: </span>In the future, the <em>population</em> must be a sequence.  Instances of\\n<a class=\"reference internal\" href=\"stdtypes.html#set\" title=\"set\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">set</span></code></a> are no longer supported.  The set must first be converted\\nto a <a class=\"reference internal\" href=\"stdtypes.html#list\" title=\"list\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">list</span></code></a> or <a class=\"reference internal\" href=\"stdtypes.html#tuple\" title=\"tuple\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">tuple</span></code></a>, preferably in a deterministic\\norder so that the sample is reproducible.</p>\\n</div>\\n</dd></dl>\\n\\n</div>\\n<div class=\"section\" id=\"real-valued-distributions\">\\n<span id=\"id1\"></span><h2>Real-valued distributions<a class=\"headerlink\" href=\"#real-valued-distributions\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<p>The following functions generate specific real-valued distributions. Function\\nparameters are named after the corresponding variables in the distribution\\xe2\\x80\\x99s\\nequation, as used in common mathematical practice; most of these equations can\\nbe found in any statistics text.</p>\\n<dl class=\"function\">\\n<dt id=\"random.random\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">random</code><span class=\"sig-paren\">(</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.random\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return the next random floating point number in the range [0.0, 1.0).</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.uniform\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">uniform</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a</em>, <em class=\"sig-param\">b</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.uniform\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return a random floating point number <em>N</em> such that <code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">b</span></code> for\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">b</span></code> and <code class=\"docutils literal notranslate\"><span class=\"pre\">b</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">a</span></code> for <code class=\"docutils literal notranslate\"><span class=\"pre\">b</span> <span class=\"pre\">&lt;</span> <span class=\"pre\">a</span></code>.</p>\\n<p>The end-point value <code class=\"docutils literal notranslate\"><span class=\"pre\">b</span></code> may or may not be included in the range\\ndepending on floating-point rounding in the equation <code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">+</span> <span class=\"pre\">(b-a)</span> <span class=\"pre\">*</span> <span class=\"pre\">random()</span></code>.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.triangular\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">triangular</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">low</em>, <em class=\"sig-param\">high</em>, <em class=\"sig-param\">mode</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.triangular\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Return a random floating point number <em>N</em> such that <code class=\"docutils literal notranslate\"><span class=\"pre\">low</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">high</span></code> and\\nwith the specified <em>mode</em> between those bounds.  The <em>low</em> and <em>high</em> bounds\\ndefault to zero and one.  The <em>mode</em> argument defaults to the midpoint\\nbetween the bounds, giving a symmetric distribution.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.betavariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">betavariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.betavariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Beta distribution.  Conditions on the parameters are <code class=\"docutils literal notranslate\"><span class=\"pre\">alpha</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code> and\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">beta</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code>. Returned values range between 0 and 1.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.expovariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">expovariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">lambd</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.expovariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Exponential distribution.  <em>lambd</em> is 1.0 divided by the desired\\nmean.  It should be nonzero.  (The parameter would be called\\n\\xe2\\x80\\x9clambda\\xe2\\x80\\x9d, but that is a reserved word in Python.)  Returned values\\nrange from 0 to positive infinity if <em>lambd</em> is positive, and from\\nnegative infinity to 0 if <em>lambd</em> is negative.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.gammavariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">gammavariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.gammavariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Gamma distribution.  (<em>Not</em> the gamma function!)  Conditions on the\\nparameters are <code class=\"docutils literal notranslate\"><span class=\"pre\">alpha</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code> and <code class=\"docutils literal notranslate\"><span class=\"pre\">beta</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code>.</p>\\n<p>The probability distribution function is:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span>          <span class=\"n\">x</span> <span class=\"o\">**</span> <span class=\"p\">(</span><span class=\"n\">alpha</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span> <span class=\"o\">*</span> <span class=\"n\">math</span><span class=\"o\">.</span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span> <span class=\"o\">/</span> <span class=\"n\">beta</span><span class=\"p\">)</span>\\n<span class=\"n\">pdf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">=</span>  <span class=\"o\">--------------------------------------</span>\\n            <span class=\"n\">math</span><span class=\"o\">.</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">alpha</span><span class=\"p\">)</span> <span class=\"o\">*</span> <span class=\"n\">beta</span> <span class=\"o\">**</span> <span class=\"n\">alpha</span>\\n</pre></div>\\n</div>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.gauss\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">gauss</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.gauss\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Gaussian distribution.  <em>mu</em> is the mean, and <em>sigma</em> is the standard\\ndeviation.  This is slightly faster than the <a class=\"reference internal\" href=\"#random.normalvariate\" title=\"random.normalvariate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">normalvariate()</span></code></a> function\\ndefined below.</p>\\n<p>Multithreading note:  When two threads call this function\\nsimultaneously, it is possible that they will receive the\\nsame return value.  This can be avoided in three ways.\\n1) Have each thread use a different instance of the random\\nnumber generator. 2) Put locks around all calls. 3) Use the\\nslower, but thread-safe <a class=\"reference internal\" href=\"#random.normalvariate\" title=\"random.normalvariate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">normalvariate()</span></code></a> function instead.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.lognormvariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">lognormvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.lognormvariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Log normal distribution.  If you take the natural logarithm of this\\ndistribution, you\\xe2\\x80\\x99ll get a normal distribution with mean <em>mu</em> and standard\\ndeviation <em>sigma</em>.  <em>mu</em> can have any value, and <em>sigma</em> must be greater than\\nzero.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.normalvariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">normalvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.normalvariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Normal distribution.  <em>mu</em> is the mean, and <em>sigma</em> is the standard deviation.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.vonmisesvariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">vonmisesvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">kappa</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.vonmisesvariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p><em>mu</em> is the mean angle, expressed in radians between 0 and 2*<em>pi</em>, and <em>kappa</em>\\nis the concentration parameter, which must be greater than or equal to zero.  If\\n<em>kappa</em> is equal to zero, this distribution reduces to a uniform random angle\\nover the range 0 to 2*<em>pi</em>.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.paretovariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">paretovariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.paretovariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Pareto distribution.  <em>alpha</em> is the shape parameter.</p>\\n</dd></dl>\\n\\n<dl class=\"function\">\\n<dt id=\"random.weibullvariate\">\\n<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">weibullvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.weibullvariate\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Weibull distribution.  <em>alpha</em> is the scale parameter and <em>beta</em> is the shape\\nparameter.</p>\\n</dd></dl>\\n\\n</div>\\n<div class=\"section\" id=\"alternative-generator\">\\n<h2>Alternative Generator<a class=\"headerlink\" href=\"#alternative-generator\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<dl class=\"class\">\\n<dt id=\"random.Random\">\\n<em class=\"property\">class </em><code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">Random</code><span class=\"sig-paren\">(</span><span class=\"optional\">[</span><em class=\"sig-param\">seed</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.Random\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Class that implements the default pseudo-random number generator used by the\\n<a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code></a> module.</p>\\n<div class=\"deprecated\">\\n<p><span class=\"versionmodified deprecated\">Deprecated since version 3.9: </span>In the future, the <em>seed</em> must be one of the following types:\\n<code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">NoneType</span></code>, <a class=\"reference internal\" href=\"functions.html#int\" title=\"int\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">int</span></code></a>, <a class=\"reference internal\" href=\"functions.html#float\" title=\"float\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">float</span></code></a>, <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a>,\\n<a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a>, or <a class=\"reference internal\" href=\"stdtypes.html#bytearray\" title=\"bytearray\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytearray</span></code></a>.</p>\\n</div>\\n</dd></dl>\\n\\n<dl class=\"class\">\\n<dt id=\"random.SystemRandom\">\\n<em class=\"property\">class </em><code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">SystemRandom</code><span class=\"sig-paren\">(</span><span class=\"optional\">[</span><em class=\"sig-param\">seed</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.SystemRandom\" title=\"Permalink to this definition\">\\xc2\\xb6</a></dt>\\n<dd><p>Class that uses the <a class=\"reference internal\" href=\"os.html#os.urandom\" title=\"os.urandom\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">os.urandom()</span></code></a> function for generating random numbers\\nfrom sources provided by the operating system. Not available on all systems.\\nDoes not rely on software state, and sequences are not reproducible. Accordingly,\\nthe <a class=\"reference internal\" href=\"#random.seed\" title=\"random.seed\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">seed()</span></code></a> method has no effect and is ignored.\\nThe <a class=\"reference internal\" href=\"#random.getstate\" title=\"random.getstate\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getstate()</span></code></a> and <a class=\"reference internal\" href=\"#random.setstate\" title=\"random.setstate\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">setstate()</span></code></a> methods raise\\n<a class=\"reference internal\" href=\"exceptions.html#NotImplementedError\" title=\"NotImplementedError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">NotImplementedError</span></code></a> if called.</p>\\n</dd></dl>\\n\\n</div>\\n<div class=\"section\" id=\"notes-on-reproducibility\">\\n<h2>Notes on Reproducibility<a class=\"headerlink\" href=\"#notes-on-reproducibility\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<p>Sometimes it is useful to be able to reproduce the sequences given by a\\npseudo-random number generator.  By re-using a seed value, the same sequence should be\\nreproducible from run to run as long as multiple threads are not running.</p>\\n<p>Most of the random module\\xe2\\x80\\x99s algorithms and seeding functions are subject to\\nchange across Python versions, but two aspects are guaranteed not to change:</p>\\n<ul class=\"simple\">\\n<li><p>If a new seeding method is added, then a backward compatible seeder will be\\noffered.</p></li>\\n<li><p>The generator\\xe2\\x80\\x99s <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">random()</span></code> method will continue to produce the same\\nsequence when the compatible seeder is given the same seed.</p></li>\\n</ul>\\n</div>\\n<div class=\"section\" id=\"examples\">\\n<span id=\"random-examples\"></span><h2>Examples<a class=\"headerlink\" href=\"#examples\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<p>Basic examples:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">random</span><span class=\"p\">()</span>                             <span class=\"c1\"># Random float:  0.0 &lt;= x &lt; 1.0</span>\\n<span class=\"go\">0.37444887175646646</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uniform</span><span class=\"p\">(</span><span class=\"mf\">2.5</span><span class=\"p\">,</span> <span class=\"mf\">10.0</span><span class=\"p\">)</span>                   <span class=\"c1\"># Random float:  2.5 &lt;= x &lt; 10.0</span>\\n<span class=\"go\">3.1800146073117523</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expovariate</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">/</span> <span class=\"mi\">5</span><span class=\"p\">)</span>                   <span class=\"c1\"># Interval between arrivals averaging 5 seconds</span>\\n<span class=\"go\">5.148957571865031</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>                        <span class=\"c1\"># Integer from 0 to 9 inclusive</span>\\n<span class=\"go\">7</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">101</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>                 <span class=\"c1\"># Even integer from 0 to 100 inclusive</span>\\n<span class=\"go\">26</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">choice</span><span class=\"p\">([</span><span class=\"s1\">&#39;win&#39;</span><span class=\"p\">,</span> <span class=\"s1\">&#39;lose&#39;</span><span class=\"p\">,</span> <span class=\"s1\">&#39;draw&#39;</span><span class=\"p\">])</span>      <span class=\"c1\"># Single random element from a sequence</span>\\n<span class=\"go\">&#39;draw&#39;</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">deck</span> <span class=\"o\">=</span> <span class=\"s1\">&#39;ace two three four&#39;</span><span class=\"o\">.</span><span class=\"n\">split</span><span class=\"p\">()</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">shuffle</span><span class=\"p\">(</span><span class=\"n\">deck</span><span class=\"p\">)</span>                        <span class=\"c1\"># Shuffle a list</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">deck</span>\\n<span class=\"go\">[&#39;four&#39;, &#39;two&#39;, &#39;ace&#39;, &#39;three&#39;]</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sample</span><span class=\"p\">([</span><span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"mi\">20</span><span class=\"p\">,</span> <span class=\"mi\">30</span><span class=\"p\">,</span> <span class=\"mi\">40</span><span class=\"p\">,</span> <span class=\"mi\">50</span><span class=\"p\">],</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>    <span class=\"c1\"># Four samples without replacement</span>\\n<span class=\"go\">[40, 10, 50, 30]</span>\\n</pre></div>\\n</div>\\n<p>Simulations:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Six roulette wheel spins (weighted sampling with replacement)</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">choices</span><span class=\"p\">([</span><span class=\"s1\">&#39;red&#39;</span><span class=\"p\">,</span> <span class=\"s1\">&#39;black&#39;</span><span class=\"p\">,</span> <span class=\"s1\">&#39;green&#39;</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">18</span><span class=\"p\">,</span> <span class=\"mi\">18</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">6</span><span class=\"p\">)</span>\\n<span class=\"go\">[&#39;red&#39;, &#39;green&#39;, &#39;black&#39;, &#39;black&#39;, &#39;red&#39;, &#39;black&#39;]</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Deal 20 cards without replacement from a deck</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># of 52 playing cards, and determine the proportion of cards</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># with a ten-value:  ten, jack, queen, or king.</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">dealt</span> <span class=\"o\">=</span> <span class=\"n\">sample</span><span class=\"p\">([</span><span class=\"s1\">&#39;tens&#39;</span><span class=\"p\">,</span> <span class=\"s1\">&#39;low cards&#39;</span><span class=\"p\">],</span> <span class=\"n\">counts</span><span class=\"o\">=</span><span class=\"p\">[</span><span class=\"mi\">16</span><span class=\"p\">,</span> <span class=\"mi\">36</span><span class=\"p\">],</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">20</span><span class=\"p\">)</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">dealt</span><span class=\"o\">.</span><span class=\"n\">count</span><span class=\"p\">(</span><span class=\"s1\">&#39;tens&#39;</span><span class=\"p\">)</span> <span class=\"o\">/</span> <span class=\"mi\">20</span>\\n<span class=\"go\">0.15</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Estimate the probability of getting 5 or more heads from 7 spins</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># of a biased coin that settles on heads 60% of the time.</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">trial</span><span class=\"p\">():</span>\\n<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"n\">choices</span><span class=\"p\">(</span><span class=\"s1\">&#39;HT&#39;</span><span class=\"p\">,</span> <span class=\"n\">cum_weights</span><span class=\"o\">=</span><span class=\"p\">(</span><span class=\"mf\">0.60</span><span class=\"p\">,</span> <span class=\"mf\">1.00</span><span class=\"p\">),</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">7</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">count</span><span class=\"p\">(</span><span class=\"s1\">&#39;H&#39;</span><span class=\"p\">)</span> <span class=\"o\">&gt;=</span> <span class=\"mi\">5</span>\\n<span class=\"gp\">...</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">sum</span><span class=\"p\">(</span><span class=\"n\">trial</span><span class=\"p\">()</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10_000</span><span class=\"p\">))</span> <span class=\"o\">/</span> <span class=\"mi\">10_000</span>\\n<span class=\"go\">0.4169</span>\\n\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Probability of the median of 5 samples being in middle two quartiles</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">trial</span><span class=\"p\">():</span>\\n<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"mi\">2_500</span> <span class=\"o\">&lt;=</span> <span class=\"nb\">sorted</span><span class=\"p\">(</span><span class=\"n\">choices</span><span class=\"p\">(</span><span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10_000</span><span class=\"p\">),</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">5</span><span class=\"p\">))[</span><span class=\"mi\">2</span><span class=\"p\">]</span> <span class=\"o\">&lt;</span> <span class=\"mi\">7_500</span>\\n<span class=\"gp\">...</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">sum</span><span class=\"p\">(</span><span class=\"n\">trial</span><span class=\"p\">()</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10_000</span><span class=\"p\">))</span> <span class=\"o\">/</span> <span class=\"mi\">10_000</span>\\n<span class=\"go\">0.7958</span>\\n</pre></div>\\n</div>\\n<p>Example of <a class=\"reference external\" href=\"https://en.wikipedia.org/wiki/Bootstrapping_(statistics)\">statistical bootstrapping</a> using resampling\\nwith replacement to estimate a confidence interval for the mean of a sample:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"c1\"># http://statistics.about.com/od/Applications/a/Example-Of-Bootstrapping.htm</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">statistics</span> <span class=\"kn\">import</span> <span class=\"n\">fmean</span> <span class=\"k\">as</span> <span class=\"n\">mean</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">choices</span>\\n\\n<span class=\"n\">data</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">41</span><span class=\"p\">,</span> <span class=\"mi\">50</span><span class=\"p\">,</span> <span class=\"mi\">29</span><span class=\"p\">,</span> <span class=\"mi\">37</span><span class=\"p\">,</span> <span class=\"mi\">81</span><span class=\"p\">,</span> <span class=\"mi\">30</span><span class=\"p\">,</span> <span class=\"mi\">73</span><span class=\"p\">,</span> <span class=\"mi\">63</span><span class=\"p\">,</span> <span class=\"mi\">20</span><span class=\"p\">,</span> <span class=\"mi\">35</span><span class=\"p\">,</span> <span class=\"mi\">68</span><span class=\"p\">,</span> <span class=\"mi\">22</span><span class=\"p\">,</span> <span class=\"mi\">60</span><span class=\"p\">,</span> <span class=\"mi\">31</span><span class=\"p\">,</span> <span class=\"mi\">95</span><span class=\"p\">]</span>\\n<span class=\"n\">means</span> <span class=\"o\">=</span> <span class=\"nb\">sorted</span><span class=\"p\">(</span><span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">choices</span><span class=\"p\">(</span><span class=\"n\">data</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">data</span><span class=\"p\">)))</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">100</span><span class=\"p\">))</span>\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;The sample mean of </span><span class=\"si\">{</span><span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">data</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\"> has a 90% confidence &#39;</span>\\n      <span class=\"sa\">f</span><span class=\"s1\">&#39;interval from </span><span class=\"si\">{</span><span class=\"n\">means</span><span class=\"p\">[</span><span class=\"mi\">5</span><span class=\"p\">]</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\"> to </span><span class=\"si\">{</span><span class=\"n\">means</span><span class=\"p\">[</span><span class=\"mi\">94</span><span class=\"p\">]</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">&#39;</span><span class=\"p\">)</span>\\n</pre></div>\\n</div>\\n<p>Example of a <a class=\"reference external\" href=\"https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests\">resampling permutation test</a>\\nto determine the statistical significance or <a class=\"reference external\" href=\"https://en.wikipedia.org/wiki/P-value\">p-value</a> of an observed difference\\nbetween the effects of a drug versus a placebo:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"c1\"># Example from &quot;Statistics is Easy&quot; by Dennis Shasha and Manda Wilson</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">statistics</span> <span class=\"kn\">import</span> <span class=\"n\">fmean</span> <span class=\"k\">as</span> <span class=\"n\">mean</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">shuffle</span>\\n\\n<span class=\"n\">drug</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">54</span><span class=\"p\">,</span> <span class=\"mi\">73</span><span class=\"p\">,</span> <span class=\"mi\">53</span><span class=\"p\">,</span> <span class=\"mi\">70</span><span class=\"p\">,</span> <span class=\"mi\">73</span><span class=\"p\">,</span> <span class=\"mi\">68</span><span class=\"p\">,</span> <span class=\"mi\">52</span><span class=\"p\">,</span> <span class=\"mi\">65</span><span class=\"p\">,</span> <span class=\"mi\">65</span><span class=\"p\">]</span>\\n<span class=\"n\">placebo</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">54</span><span class=\"p\">,</span> <span class=\"mi\">51</span><span class=\"p\">,</span> <span class=\"mi\">58</span><span class=\"p\">,</span> <span class=\"mi\">44</span><span class=\"p\">,</span> <span class=\"mi\">55</span><span class=\"p\">,</span> <span class=\"mi\">52</span><span class=\"p\">,</span> <span class=\"mi\">42</span><span class=\"p\">,</span> <span class=\"mi\">47</span><span class=\"p\">,</span> <span class=\"mi\">58</span><span class=\"p\">,</span> <span class=\"mi\">46</span><span class=\"p\">]</span>\\n<span class=\"n\">observed_diff</span> <span class=\"o\">=</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">drug</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">placebo</span><span class=\"p\">)</span>\\n\\n<span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"mi\">10_000</span>\\n<span class=\"n\">count</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\\n<span class=\"n\">combined</span> <span class=\"o\">=</span> <span class=\"n\">drug</span> <span class=\"o\">+</span> <span class=\"n\">placebo</span>\\n<span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">):</span>\\n    <span class=\"n\">shuffle</span><span class=\"p\">(</span><span class=\"n\">combined</span><span class=\"p\">)</span>\\n    <span class=\"n\">new_diff</span> <span class=\"o\">=</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">combined</span><span class=\"p\">[:</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">drug</span><span class=\"p\">)])</span> <span class=\"o\">-</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">combined</span><span class=\"p\">[</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">drug</span><span class=\"p\">):])</span>\\n    <span class=\"n\">count</span> <span class=\"o\">+=</span> <span class=\"p\">(</span><span class=\"n\">new_diff</span> <span class=\"o\">&gt;=</span> <span class=\"n\">observed_diff</span><span class=\"p\">)</span>\\n\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;</span><span class=\"si\">{</span><span class=\"n\">n</span><span class=\"si\">}</span><span class=\"s1\"> label reshufflings produced only </span><span class=\"si\">{</span><span class=\"n\">count</span><span class=\"si\">}</span><span class=\"s1\"> instances with a difference&#39;</span><span class=\"p\">)</span>\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;at least as extreme as the observed difference of </span><span class=\"si\">{</span><span class=\"n\">observed_diff</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.&#39;</span><span class=\"p\">)</span>\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;The one-sided p-value of </span><span class=\"si\">{</span><span class=\"n\">count</span> <span class=\"o\">/</span> <span class=\"n\">n</span><span class=\"si\">:</span><span class=\"s1\">.4f</span><span class=\"si\">}</span><span class=\"s1\"> leads us to reject the null&#39;</span><span class=\"p\">)</span>\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;hypothesis that there is no difference between the drug and the placebo.&#39;</span><span class=\"p\">)</span>\\n</pre></div>\\n</div>\\n<p>Simulation of arrival times and service deliveries for a multiserver queue:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"kn\">from</span> <span class=\"nn\">heapq</span> <span class=\"kn\">import</span> <span class=\"n\">heappush</span><span class=\"p\">,</span> <span class=\"n\">heappop</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">expovariate</span><span class=\"p\">,</span> <span class=\"n\">gauss</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">statistics</span> <span class=\"kn\">import</span> <span class=\"n\">mean</span><span class=\"p\">,</span> <span class=\"n\">median</span><span class=\"p\">,</span> <span class=\"n\">stdev</span>\\n\\n<span class=\"n\">average_arrival_interval</span> <span class=\"o\">=</span> <span class=\"mf\">5.6</span>\\n<span class=\"n\">average_service_time</span> <span class=\"o\">=</span> <span class=\"mf\">15.0</span>\\n<span class=\"n\">stdev_service_time</span> <span class=\"o\">=</span> <span class=\"mf\">3.5</span>\\n<span class=\"n\">num_servers</span> <span class=\"o\">=</span> <span class=\"mi\">3</span>\\n\\n<span class=\"n\">waits</span> <span class=\"o\">=</span> <span class=\"p\">[]</span>\\n<span class=\"n\">arrival_time</span> <span class=\"o\">=</span> <span class=\"mf\">0.0</span>\\n<span class=\"n\">servers</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mf\">0.0</span><span class=\"p\">]</span> <span class=\"o\">*</span> <span class=\"n\">num_servers</span>  <span class=\"c1\"># time when each server becomes available</span>\\n<span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">100_000</span><span class=\"p\">):</span>\\n    <span class=\"n\">arrival_time</span> <span class=\"o\">+=</span> <span class=\"n\">expovariate</span><span class=\"p\">(</span><span class=\"mf\">1.0</span> <span class=\"o\">/</span> <span class=\"n\">average_arrival_interval</span><span class=\"p\">)</span>\\n    <span class=\"n\">next_server_available</span> <span class=\"o\">=</span> <span class=\"n\">heappop</span><span class=\"p\">(</span><span class=\"n\">servers</span><span class=\"p\">)</span>\\n    <span class=\"n\">wait</span> <span class=\"o\">=</span> <span class=\"nb\">max</span><span class=\"p\">(</span><span class=\"mf\">0.0</span><span class=\"p\">,</span> <span class=\"n\">next_server_available</span> <span class=\"o\">-</span> <span class=\"n\">arrival_time</span><span class=\"p\">)</span>\\n    <span class=\"n\">waits</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">wait</span><span class=\"p\">)</span>\\n    <span class=\"n\">service_duration</span> <span class=\"o\">=</span> <span class=\"n\">gauss</span><span class=\"p\">(</span><span class=\"n\">average_service_time</span><span class=\"p\">,</span> <span class=\"n\">stdev_service_time</span><span class=\"p\">)</span>\\n    <span class=\"n\">service_completed</span> <span class=\"o\">=</span> <span class=\"n\">arrival_time</span> <span class=\"o\">+</span> <span class=\"n\">wait</span> <span class=\"o\">+</span> <span class=\"n\">service_duration</span>\\n    <span class=\"n\">heappush</span><span class=\"p\">(</span><span class=\"n\">servers</span><span class=\"p\">,</span> <span class=\"n\">service_completed</span><span class=\"p\">)</span>\\n\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;Mean wait: </span><span class=\"si\">{</span><span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.  Stdev wait: </span><span class=\"si\">{</span><span class=\"n\">stdev</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.&#39;</span><span class=\"p\">)</span>\\n<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">&#39;Median wait: </span><span class=\"si\">{</span><span class=\"n\">median</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.  Max wait: </span><span class=\"si\">{</span><span class=\"nb\">max</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.&#39;</span><span class=\"p\">)</span>\\n</pre></div>\\n</div>\\n<div class=\"admonition seealso\">\\n<p class=\"admonition-title\">See also</p>\\n<p><a class=\"reference external\" href=\"https://www.youtube.com/watch?v=Iq9DzN6mvYA\">Statistics for Hackers</a>\\na video tutorial by\\n<a class=\"reference external\" href=\"https://us.pycon.org/2016/speaker/profile/295/\">Jake Vanderplas</a>\\non statistical analysis using just a few fundamental concepts\\nincluding simulation, sampling, shuffling, and cross-validation.</p>\\n<p><a class=\"reference external\" href=\"http://nbviewer.jupyter.org/url/norvig.com/ipython/Economics.ipynb\">Economics Simulation</a>\\na simulation of a marketplace by\\n<a class=\"reference external\" href=\"http://norvig.com/bio.html\">Peter Norvig</a> that shows effective\\nuse of many of the tools and distributions provided by this module\\n(gauss, uniform, sample, betavariate, choice, triangular, and randrange).</p>\\n<p><a class=\"reference external\" href=\"http://nbviewer.jupyter.org/url/norvig.com/ipython/Probability.ipynb\">A Concrete Introduction to Probability (using Python)</a>\\na tutorial by <a class=\"reference external\" href=\"http://norvig.com/bio.html\">Peter Norvig</a> covering\\nthe basics of probability theory, how to write simulations, and\\nhow to perform data analysis using Python.</p>\\n</div>\\n</div>\\n<div class=\"section\" id=\"recipes\">\\n<h2>Recipes<a class=\"headerlink\" href=\"#recipes\" title=\"Permalink to this headline\">\\xc2\\xb6</a></h2>\\n<p>The default <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a> returns multiples of 2\\xe2\\x81\\xbb\\xe2\\x81\\xb5\\xc2\\xb3 in the range\\n<em>0.0 \\xe2\\x89\\xa4 x &lt; 1.0</em>.  All such numbers are evenly spaced and are exactly\\nrepresentable as Python floats.  However, many other representable\\nfloats in that interval are not possible selections.  For example,\\n<code class=\"docutils literal notranslate\"><span class=\"pre\">0.05954861408025609</span></code> isn\\xe2\\x80\\x99t an integer multiple of 2\\xe2\\x81\\xbb\\xe2\\x81\\xb5\\xc2\\xb3.</p>\\n<p>The following recipe takes a different approach.  All floats in the\\ninterval are possible selections.  The mantissa comes from a uniform\\ndistribution of integers in the range <em>2\\xe2\\x81\\xb5\\xc2\\xb2 \\xe2\\x89\\xa4 mantissa &lt; 2\\xe2\\x81\\xb5\\xc2\\xb3</em>.  The\\nexponent comes from a geometric distribution where exponents smaller\\nthan <em>-53</em> occur half as often as the next larger exponent.</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">Random</span>\\n<span class=\"kn\">from</span> <span class=\"nn\">math</span> <span class=\"kn\">import</span> <span class=\"n\">ldexp</span>\\n\\n<span class=\"k\">class</span> <span class=\"nc\">FullRandom</span><span class=\"p\">(</span><span class=\"n\">Random</span><span class=\"p\">):</span>\\n\\n    <span class=\"k\">def</span> <span class=\"nf\">random</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">):</span>\\n        <span class=\"n\">mantissa</span> <span class=\"o\">=</span> <span class=\"mh\">0x10_0000_0000_0000</span> <span class=\"o\">|</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">getrandbits</span><span class=\"p\">(</span><span class=\"mi\">52</span><span class=\"p\">)</span>\\n        <span class=\"n\">exponent</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"mi\">53</span>\\n        <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\\n        <span class=\"k\">while</span> <span class=\"ow\">not</span> <span class=\"n\">x</span><span class=\"p\">:</span>\\n            <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">getrandbits</span><span class=\"p\">(</span><span class=\"mi\">32</span><span class=\"p\">)</span>\\n            <span class=\"n\">exponent</span> <span class=\"o\">+=</span> <span class=\"n\">x</span><span class=\"o\">.</span><span class=\"n\">bit_length</span><span class=\"p\">()</span> <span class=\"o\">-</span> <span class=\"mi\">32</span>\\n        <span class=\"k\">return</span> <span class=\"n\">ldexp</span><span class=\"p\">(</span><span class=\"n\">mantissa</span><span class=\"p\">,</span> <span class=\"n\">exponent</span><span class=\"p\">)</span>\\n</pre></div>\\n</div>\\n<p>All <a class=\"reference internal\" href=\"#real-valued-distributions\"><span class=\"std std-ref\">real valued distributions</span></a>\\nin the class will use the new method:</p>\\n<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fr</span> <span class=\"o\">=</span> <span class=\"n\">FullRandom</span><span class=\"p\">()</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fr</span><span class=\"o\">.</span><span class=\"n\">random</span><span class=\"p\">()</span>\\n<span class=\"go\">0.05954861408025609</span>\\n<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fr</span><span class=\"o\">.</span><span class=\"n\">expovariate</span><span class=\"p\">(</span><span class=\"mf\">0.25</span><span class=\"p\">)</span>\\n<span class=\"go\">8.87925541791544</span>\\n</pre></div>\\n</div>\\n<p>The recipe is conceptually equivalent to an algorithm that chooses from\\nall the multiples of 2\\xe2\\x81\\xbb\\xc2\\xb9\\xe2\\x81\\xb0\\xe2\\x81\\xb7\\xe2\\x81\\xb4 in the range <em>0.0 \\xe2\\x89\\xa4 x &lt; 1.0</em>.  All such\\nnumbers are evenly spaced, but most have to be rounded down to the\\nnearest representable Python float.  (The value 2\\xe2\\x81\\xbb\\xc2\\xb9\\xe2\\x81\\xb0\\xe2\\x81\\xb7\\xe2\\x81\\xb4 is the smallest\\npositive unnormalized float and is equal to <code class=\"docutils literal notranslate\"><span class=\"pre\">math.ulp(0.0)</span></code>.)</p>\\n<div class=\"admonition seealso\">\\n<p class=\"admonition-title\">See also</p>\\n<p><a class=\"reference external\" href=\"https://allendowney.com/research/rand/downey07randfloat.pdf\">Generating Pseudo-random Floating-Point Values</a> a\\npaper by Allen B. Downey describing ways to generate more\\nfine-grained floats than normally generated by <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a>.</p>\\n</div>\\n</div>\\n</div>\\n\\n\\n          </div>\\n        </div>\\n      </div>\\n      <div class=\"sphinxsidebar\" role=\"navigation\" aria-label=\"main navigation\">\\n        <div class=\"sphinxsidebarwrapper\">\\n  <h3><a href=\"../contents.html\">Table of Contents</a></h3>\\n  <ul>\\n<li><a class=\"reference internal\" href=\"#\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code> \\xe2\\x80\\x94 Generate pseudo-random numbers</a><ul>\\n<li><a class=\"reference internal\" href=\"#bookkeeping-functions\">Bookkeeping functions</a></li>\\n<li><a class=\"reference internal\" href=\"#functions-for-bytes\">Functions for bytes</a></li>\\n<li><a class=\"reference internal\" href=\"#functions-for-integers\">Functions for integers</a></li>\\n<li><a class=\"reference internal\" href=\"#functions-for-sequences\">Functions for sequences</a></li>\\n<li><a class=\"reference internal\" href=\"#real-valued-distributions\">Real-valued distributions</a></li>\\n<li><a class=\"reference internal\" href=\"#alternative-generator\">Alternative Generator</a></li>\\n<li><a class=\"reference internal\" href=\"#notes-on-reproducibility\">Notes on Reproducibility</a></li>\\n<li><a class=\"reference internal\" href=\"#examples\">Examples</a></li>\\n<li><a class=\"reference internal\" href=\"#recipes\">Recipes</a></li>\\n</ul>\\n</li>\\n</ul>\\n\\n  <h4>Previous topic</h4>\\n  <p class=\"topless\"><a href=\"fractions.html\"\\n                        title=\"previous chapter\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">fractions</span></code> \\xe2\\x80\\x94 Rational numbers</a></p>\\n  <h4>Next topic</h4>\\n  <p class=\"topless\"><a href=\"statistics.html\"\\n                        title=\"next chapter\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">statistics</span></code> \\xe2\\x80\\x94 Mathematical statistics functions</a></p>\\n  <div role=\"note\" aria-label=\"source link\">\\n    <h3>This Page</h3>\\n    <ul class=\"this-page-menu\">\\n      <li><a href=\"../bugs.html\">Report a Bug</a></li>\\n      <li>\\n        <a href=\"https://github.com/python/cpython/blob/master/Doc/library/random.rst\"\\n            rel=\"nofollow\">Show Source\\n        </a>\\n      </li>\\n    </ul>\\n  </div>\\n        </div>\\n      </div>\\n      <div class=\"clearer\"></div>\\n    </div>  \\n    <div class=\"related\" role=\"navigation\" aria-label=\"related navigation\">\\n      <h3>Navigation</h3>\\n      <ul>\\n        <li class=\"right\" style=\"margin-right: 10px\">\\n          <a href=\"../genindex.html\" title=\"General Index\"\\n             >index</a></li>\\n        <li class=\"right\" >\\n          <a href=\"../py-modindex.html\" title=\"Python Module Index\"\\n             >modules</a> |</li>\\n        <li class=\"right\" >\\n          <a href=\"statistics.html\" title=\"statistics \\xe2\\x80\\x94 Mathematical statistics functions\"\\n             >next</a> |</li>\\n        <li class=\"right\" >\\n          <a href=\"fractions.html\" title=\"fractions \\xe2\\x80\\x94 Rational numbers\"\\n             >previous</a> |</li>\\n\\n    <li><img src=\"../_static/py.png\" alt=\"\"\\n             style=\"vertical-align: middle; margin-top: -1px\"/></li>\\n    <li><a href=\"https://www.python.org/\">Python</a> &#187;</li>\\n    \\n\\n    <li>\\n      <a href=\"../index.html\">3.9.2 Documentation</a> &#187;\\n    </li>\\n\\n          <li class=\"nav-item nav-item-1\"><a href=\"index.html\" >The Python Standard Library</a> &#187;</li>\\n          <li class=\"nav-item nav-item-2\"><a href=\"numeric.html\" >Numeric and Mathematical Modules</a> &#187;</li>\\n    <li class=\"right\">\\n        \\n\\n    <div class=\"inline-search\" style=\"display: none\" role=\"search\">\\n        <form class=\"inline-search\" action=\"../search.html\" method=\"get\">\\n          <input placeholder=\"Quick search\" type=\"text\" name=\"q\" />\\n          <input type=\"submit\" value=\"Go\" />\\n          <input type=\"hidden\" name=\"check_keywords\" value=\"yes\" />\\n          <input type=\"hidden\" name=\"area\" value=\"default\" />\\n        </form>\\n    </div>\\n    <script type=\"text/javascript\">$(\\'.inline-search\\').show(0);</script>\\n         |\\n    </li>\\n\\n      </ul>\\n    </div>  \\n    <div class=\"footer\">\\n    &copy; <a href=\"../copyright.html\">Copyright</a> 2001-2021, Python Software Foundation.\\n    <br />\\n\\n    The Python Software Foundation is a non-profit corporation.\\n<a href=\"https://www.python.org/psf/donations/\">Please donate.</a>\\n<br />\\n    <br />\\n\\n    Last updated on Mar 24, 2021.\\n    <a href=\"https://docs.python.org/3/bugs.html\">Found a bug</a>?\\n    <br />\\n\\n    Created using <a href=\"https://www.sphinx-doc.org/\">Sphinx</a> 2.4.4.\\n    </div>\\n\\n    <script type=\"text/javascript\" src=\"../_static/switchers.js\"></script>\\n  </body>\\n</html>'"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "response.content"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a lot to look at! It's also pretty unreadable. This is where Beautiful Soup comes in. What Beautiful Soup does is helps us parse the page content properly, into a form that we can more easily use."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "bs4.BeautifulSoup"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Turn the undecoded content into a Beautiful Soup object and assign it to a variable\n",
    "soup = BeautifulSoup(response.content)\n",
    "type(soup)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Now let's take a look at this.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "scrolled": true,
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
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       "<div class=\"section\" id=\"module-random\">\n",
       "<span id=\"random-generate-pseudo-random-numbers\"></span><h1><a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code></a> — Generate pseudo-random numbers<a class=\"headerlink\" href=\"#module-random\" title=\"Permalink to this headline\">¶</a></h1>\n",
       "<p><strong>Source code:</strong> <a class=\"reference external\" href=\"https://github.com/python/cpython/tree/3.9/Lib/random.py\">Lib/random.py</a></p>\n",
       "<hr class=\"docutils\"/>\n",
       "<p>This module implements pseudo-random number generators for various\n",
       "distributions.</p>\n",
       "<p>For integers, there is uniform selection from a range. For sequences, there is\n",
       "uniform selection of a random element, a function to generate a random\n",
       "permutation of a list in-place, and a function for random sampling without\n",
       "replacement.</p>\n",
       "<p>On the real line, there are functions to compute uniform, normal (Gaussian),\n",
       "lognormal, negative exponential, gamma, and beta distributions. For generating\n",
       "distributions of angles, the von Mises distribution is available.</p>\n",
       "<p>Almost all module functions depend on the basic function <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a>, which\n",
       "generates a random float uniformly in the semi-open range [0.0, 1.0).  Python\n",
       "uses the Mersenne Twister as the core generator.  It produces 53-bit precision\n",
       "floats and has a period of 2**19937-1.  The underlying implementation in C is\n",
       "both fast and threadsafe.  The Mersenne Twister is one of the most extensively\n",
       "tested random number generators in existence.  However, being completely\n",
       "deterministic, it is not suitable for all purposes, and is completely unsuitable\n",
       "for cryptographic purposes.</p>\n",
       "<p>The functions supplied by this module are actually bound methods of a hidden\n",
       "instance of the <a class=\"reference internal\" href=\"#random.Random\" title=\"random.Random\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">random.Random</span></code></a> class.  You can instantiate your own\n",
       "instances of <a class=\"reference internal\" href=\"#random.Random\" title=\"random.Random\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">Random</span></code></a> to get generators that don’t share state.</p>\n",
       "<p>Class <a class=\"reference internal\" href=\"#random.Random\" title=\"random.Random\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">Random</span></code></a> can also be subclassed if you want to use a different\n",
       "basic generator of your own devising: in that case, override the <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">random()</span></code>,\n",
       "<code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">seed()</span></code>, <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getstate()</span></code>, and <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">setstate()</span></code> methods.\n",
       "Optionally, a new generator can supply a <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getrandbits()</span></code> method — this\n",
       "allows <a class=\"reference internal\" href=\"#random.randrange\" title=\"random.randrange\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">randrange()</span></code></a> to produce selections over an arbitrarily large range.</p>\n",
       "<p>The <a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code></a> module also provides the <a class=\"reference internal\" href=\"#random.SystemRandom\" title=\"random.SystemRandom\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">SystemRandom</span></code></a> class which\n",
       "uses the system function <a class=\"reference internal\" href=\"os.html#os.urandom\" title=\"os.urandom\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">os.urandom()</span></code></a> to generate random numbers\n",
       "from sources provided by the operating system.</p>\n",
       "<div class=\"admonition warning\">\n",
       "<p class=\"admonition-title\">Warning</p>\n",
       "<p>The pseudo-random generators of this module should not be used for\n",
       "security purposes.  For security or cryptographic uses, see the\n",
       "<a class=\"reference internal\" href=\"secrets.html#module-secrets\" title=\"secrets: Generate secure random numbers for managing secrets.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">secrets</span></code></a> module.</p>\n",
       "</div>\n",
       "<div class=\"admonition seealso\">\n",
       "<p class=\"admonition-title\">See also</p>\n",
       "<p>M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally\n",
       "equidistributed uniform pseudorandom number generator”, ACM Transactions on\n",
       "Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.</p>\n",
       "<p><a class=\"reference external\" href=\"https://code.activestate.com/recipes/576707/\">Complementary-Multiply-with-Carry recipe</a> for a compatible alternative\n",
       "random number generator with a long period and comparatively simple update\n",
       "operations.</p>\n",
       "</div>\n",
       "<div class=\"section\" id=\"bookkeeping-functions\">\n",
       "<h2>Bookkeeping functions<a class=\"headerlink\" href=\"#bookkeeping-functions\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.seed\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">seed</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a=None</em>, <em class=\"sig-param\">version=2</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.seed\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Initialize the random number generator.</p>\n",
       "<p>If <em>a</em> is omitted or <code class=\"docutils literal notranslate\"><span class=\"pre\">None</span></code>, the current system time is used.  If\n",
       "randomness sources are provided by the operating system, they are used\n",
       "instead of the system time (see the <a class=\"reference internal\" href=\"os.html#os.urandom\" title=\"os.urandom\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">os.urandom()</span></code></a> function for details\n",
       "on availability).</p>\n",
       "<p>If <em>a</em> is an int, it is used directly.</p>\n",
       "<p>With version 2 (the default), a <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a>, <a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a>, or <a class=\"reference internal\" href=\"stdtypes.html#bytearray\" title=\"bytearray\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytearray</span></code></a>\n",
       "object gets converted to an <a class=\"reference internal\" href=\"functions.html#int\" title=\"int\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">int</span></code></a> and all of its bits are used.</p>\n",
       "<p>With version 1 (provided for reproducing random sequences from older versions\n",
       "of Python), the algorithm for <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a> and <a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a> generates a\n",
       "narrower range of seeds.</p>\n",
       "<div class=\"versionchanged\">\n",
       "<p><span class=\"versionmodified changed\">Changed in version 3.2: </span>Moved to the version 2 scheme which uses all of the bits in a string seed.</p>\n",
       "</div>\n",
       "<div class=\"deprecated\">\n",
       "<p><span class=\"versionmodified deprecated\">Deprecated since version 3.9: </span>In the future, the <em>seed</em> must be one of the following types:\n",
       "<em>NoneType</em>, <a class=\"reference internal\" href=\"functions.html#int\" title=\"int\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">int</span></code></a>, <a class=\"reference internal\" href=\"functions.html#float\" title=\"float\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">float</span></code></a>, <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a>,\n",
       "<a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a>, or <a class=\"reference internal\" href=\"stdtypes.html#bytearray\" title=\"bytearray\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytearray</span></code></a>.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.getstate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">getstate</code><span class=\"sig-paren\">(</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.getstate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return an object capturing the current internal state of the generator.  This\n",
       "object can be passed to <a class=\"reference internal\" href=\"#random.setstate\" title=\"random.setstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">setstate()</span></code></a> to restore the state.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.setstate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">setstate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">state</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.setstate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p><em>state</em> should have been obtained from a previous call to <a class=\"reference internal\" href=\"#random.getstate\" title=\"random.getstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">getstate()</span></code></a>, and\n",
       "<a class=\"reference internal\" href=\"#random.setstate\" title=\"random.setstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">setstate()</span></code></a> restores the internal state of the generator to what it was at\n",
       "the time <a class=\"reference internal\" href=\"#random.getstate\" title=\"random.getstate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">getstate()</span></code></a> was called.</p>\n",
       "</dd></dl>\n",
       "</div>\n",
       "<div class=\"section\" id=\"functions-for-bytes\">\n",
       "<h2>Functions for bytes<a class=\"headerlink\" href=\"#functions-for-bytes\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.randbytes\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randbytes</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">n</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randbytes\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Generate <em>n</em> random bytes.</p>\n",
       "<p>This method should not be used for generating security tokens.\n",
       "Use <a class=\"reference internal\" href=\"secrets.html#secrets.token_bytes\" title=\"secrets.token_bytes\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">secrets.token_bytes()</span></code></a> instead.</p>\n",
       "<div class=\"versionadded\">\n",
       "<p><span class=\"versionmodified added\">New in version 3.9.</span></p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "</div>\n",
       "<div class=\"section\" id=\"functions-for-integers\">\n",
       "<h2>Functions for integers<a class=\"headerlink\" href=\"#functions-for-integers\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.randrange\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randrange</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">stop</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randrange\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dt>\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randrange</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">start</em>, <em class=\"sig-param\">stop</em><span class=\"optional\">[</span>, <em class=\"sig-param\">step</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span></dt>\n",
       "<dd><p>Return a randomly selected element from <code class=\"docutils literal notranslate\"><span class=\"pre\">range(start,</span> <span class=\"pre\">stop,</span> <span class=\"pre\">step)</span></code>.  This is\n",
       "equivalent to <code class=\"docutils literal notranslate\"><span class=\"pre\">choice(range(start,</span> <span class=\"pre\">stop,</span> <span class=\"pre\">step))</span></code>, but doesn’t actually build a\n",
       "range object.</p>\n",
       "<p>The positional argument pattern matches that of <a class=\"reference internal\" href=\"stdtypes.html#range\" title=\"range\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">range()</span></code></a>.  Keyword arguments\n",
       "should not be used because the function may use them in unexpected ways.</p>\n",
       "<div class=\"versionchanged\">\n",
       "<p><span class=\"versionmodified changed\">Changed in version 3.2: </span><a class=\"reference internal\" href=\"#random.randrange\" title=\"random.randrange\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">randrange()</span></code></a> is more sophisticated about producing equally distributed\n",
       "values.  Formerly it used a style like <code class=\"docutils literal notranslate\"><span class=\"pre\">int(random()*n)</span></code> which could produce\n",
       "slightly uneven distributions.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.randint\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randint</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a</em>, <em class=\"sig-param\">b</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randint\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return a random integer <em>N</em> such that <code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">b</span></code>.  Alias for\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">randrange(a,</span> <span class=\"pre\">b+1)</span></code>.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.getrandbits\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">getrandbits</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">k</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.getrandbits\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Returns a non-negative Python integer with <em>k</em> random bits. This method\n",
       "is supplied with the MersenneTwister generator and some other generators\n",
       "may also provide it as an optional part of the API. When available,\n",
       "<a class=\"reference internal\" href=\"#random.getrandbits\" title=\"random.getrandbits\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getrandbits()</span></code></a> enables <a class=\"reference internal\" href=\"#random.randrange\" title=\"random.randrange\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">randrange()</span></code></a> to handle arbitrarily large\n",
       "ranges.</p>\n",
       "<div class=\"versionchanged\">\n",
       "<p><span class=\"versionmodified changed\">Changed in version 3.9: </span>This method now accepts zero for <em>k</em>.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "</div>\n",
       "<div class=\"section\" id=\"functions-for-sequences\">\n",
       "<h2>Functions for sequences<a class=\"headerlink\" href=\"#functions-for-sequences\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.choice\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">choice</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">seq</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.choice\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return a random element from the non-empty sequence <em>seq</em>. If <em>seq</em> is empty,\n",
       "raises <a class=\"reference internal\" href=\"exceptions.html#IndexError\" title=\"IndexError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">IndexError</span></code></a>.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.choices\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">choices</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">population</em>, <em class=\"sig-param\">weights=None</em>, <em class=\"sig-param\">*</em>, <em class=\"sig-param\">cum_weights=None</em>, <em class=\"sig-param\">k=1</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.choices\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return a <em>k</em> sized list of elements chosen from the <em>population</em> with replacement.\n",
       "If the <em>population</em> is empty, raises <a class=\"reference internal\" href=\"exceptions.html#IndexError\" title=\"IndexError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">IndexError</span></code></a>.</p>\n",
       "<p>If a <em>weights</em> sequence is specified, selections are made according to the\n",
       "relative weights.  Alternatively, if a <em>cum_weights</em> sequence is given, the\n",
       "selections are made according to the cumulative weights (perhaps computed\n",
       "using <a class=\"reference internal\" href=\"itertools.html#itertools.accumulate\" title=\"itertools.accumulate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">itertools.accumulate()</span></code></a>).  For example, the relative weights\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">[10,</span> <span class=\"pre\">5,</span> <span class=\"pre\">30,</span> <span class=\"pre\">5]</span></code> are equivalent to the cumulative weights\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">[10,</span> <span class=\"pre\">15,</span> <span class=\"pre\">45,</span> <span class=\"pre\">50]</span></code>.  Internally, the relative weights are converted to\n",
       "cumulative weights before making selections, so supplying the cumulative\n",
       "weights saves work.</p>\n",
       "<p>If neither <em>weights</em> nor <em>cum_weights</em> are specified, selections are made\n",
       "with equal probability.  If a weights sequence is supplied, it must be\n",
       "the same length as the <em>population</em> sequence.  It is a <a class=\"reference internal\" href=\"exceptions.html#TypeError\" title=\"TypeError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">TypeError</span></code></a>\n",
       "to specify both <em>weights</em> and <em>cum_weights</em>.</p>\n",
       "<p>The <em>weights</em> or <em>cum_weights</em> can use any numeric type that interoperates\n",
       "with the <a class=\"reference internal\" href=\"functions.html#float\" title=\"float\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">float</span></code></a> values returned by <a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a> (that includes\n",
       "integers, floats, and fractions but excludes decimals).  Behavior is\n",
       "undefined if any weight is negative.  A <a class=\"reference internal\" href=\"exceptions.html#ValueError\" title=\"ValueError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">ValueError</span></code></a> is raised if all\n",
       "weights are zero.</p>\n",
       "<p>For a given seed, the <a class=\"reference internal\" href=\"#random.choices\" title=\"random.choices\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choices()</span></code></a> function with equal weighting\n",
       "typically produces a different sequence than repeated calls to\n",
       "<a class=\"reference internal\" href=\"#random.choice\" title=\"random.choice\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choice()</span></code></a>.  The algorithm used by <a class=\"reference internal\" href=\"#random.choices\" title=\"random.choices\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choices()</span></code></a> uses floating\n",
       "point arithmetic for internal consistency and speed.  The algorithm used\n",
       "by <a class=\"reference internal\" href=\"#random.choice\" title=\"random.choice\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">choice()</span></code></a> defaults to integer arithmetic with repeated selections\n",
       "to avoid small biases from round-off error.</p>\n",
       "<div class=\"versionadded\">\n",
       "<p><span class=\"versionmodified added\">New in version 3.6.</span></p>\n",
       "</div>\n",
       "<div class=\"versionchanged\">\n",
       "<p><span class=\"versionmodified changed\">Changed in version 3.9: </span>Raises a <a class=\"reference internal\" href=\"exceptions.html#ValueError\" title=\"ValueError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">ValueError</span></code></a> if all weights are zero.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.shuffle\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">shuffle</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">x</em><span class=\"optional\">[</span>, <em class=\"sig-param\">random</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.shuffle\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Shuffle the sequence <em>x</em> in place.</p>\n",
       "<p>The optional argument <em>random</em> is a 0-argument function returning a random\n",
       "float in [0.0, 1.0); by default, this is the function <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a>.</p>\n",
       "<p>To shuffle an immutable sequence and return a new shuffled list, use\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">sample(x,</span> <span class=\"pre\">k=len(x))</span></code> instead.</p>\n",
       "<p>Note that even for small <code class=\"docutils literal notranslate\"><span class=\"pre\">len(x)</span></code>, the total number of permutations of <em>x</em>\n",
       "can quickly grow larger than the period of most random number generators.\n",
       "This implies that most permutations of a long sequence can never be\n",
       "generated.  For example, a sequence of length 2080 is the largest that\n",
       "can fit within the period of the Mersenne Twister random number generator.</p>\n",
       "<div class=\"deprecated-removed\">\n",
       "<p><span class=\"versionmodified\">Deprecated since version 3.9, will be removed in version 3.11: </span>The optional parameter <em>random</em>.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.sample\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">sample</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">population</em>, <em class=\"sig-param\">k</em>, <em class=\"sig-param\">*</em>, <em class=\"sig-param\">counts=None</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.sample\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return a <em>k</em> length list of unique elements chosen from the population sequence\n",
       "or set. Used for random sampling without replacement.</p>\n",
       "<p>Returns a new list containing elements from the population while leaving the\n",
       "original population unchanged.  The resulting list is in selection order so that\n",
       "all sub-slices will also be valid random samples.  This allows raffle winners\n",
       "(the sample) to be partitioned into grand prize and second place winners (the\n",
       "subslices).</p>\n",
       "<p>Members of the population need not be <a class=\"reference internal\" href=\"../glossary.html#term-hashable\"><span class=\"xref std std-term\">hashable</span></a> or unique.  If the population\n",
       "contains repeats, then each occurrence is a possible selection in the sample.</p>\n",
       "<p>Repeated elements can be specified one at a time or with the optional\n",
       "keyword-only <em>counts</em> parameter.  For example, <code class=\"docutils literal notranslate\"><span class=\"pre\">sample(['red',</span> <span class=\"pre\">'blue'],</span>\n",
       "<span class=\"pre\">counts=[4,</span> <span class=\"pre\">2],</span> <span class=\"pre\">k=5)</span></code> is equivalent to <code class=\"docutils literal notranslate\"><span class=\"pre\">sample(['red',</span> <span class=\"pre\">'red',</span> <span class=\"pre\">'red',</span> <span class=\"pre\">'red',</span>\n",
       "<span class=\"pre\">'blue',</span> <span class=\"pre\">'blue'],</span> <span class=\"pre\">k=5)</span></code>.</p>\n",
       "<p>To choose a sample from a range of integers, use a <a class=\"reference internal\" href=\"stdtypes.html#range\" title=\"range\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">range()</span></code></a> object as an\n",
       "argument.  This is especially fast and space efficient for sampling from a large\n",
       "population:  <code class=\"docutils literal notranslate\"><span class=\"pre\">sample(range(10000000),</span> <span class=\"pre\">k=60)</span></code>.</p>\n",
       "<p>If the sample size is larger than the population size, a <a class=\"reference internal\" href=\"exceptions.html#ValueError\" title=\"ValueError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">ValueError</span></code></a>\n",
       "is raised.</p>\n",
       "<div class=\"versionchanged\">\n",
       "<p><span class=\"versionmodified changed\">Changed in version 3.9: </span>Added the <em>counts</em> parameter.</p>\n",
       "</div>\n",
       "<div class=\"deprecated\">\n",
       "<p><span class=\"versionmodified deprecated\">Deprecated since version 3.9: </span>In the future, the <em>population</em> must be a sequence.  Instances of\n",
       "<a class=\"reference internal\" href=\"stdtypes.html#set\" title=\"set\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">set</span></code></a> are no longer supported.  The set must first be converted\n",
       "to a <a class=\"reference internal\" href=\"stdtypes.html#list\" title=\"list\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">list</span></code></a> or <a class=\"reference internal\" href=\"stdtypes.html#tuple\" title=\"tuple\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">tuple</span></code></a>, preferably in a deterministic\n",
       "order so that the sample is reproducible.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "</div>\n",
       "<div class=\"section\" id=\"real-valued-distributions\">\n",
       "<span id=\"id1\"></span><h2>Real-valued distributions<a class=\"headerlink\" href=\"#real-valued-distributions\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<p>The following functions generate specific real-valued distributions. Function\n",
       "parameters are named after the corresponding variables in the distribution’s\n",
       "equation, as used in common mathematical practice; most of these equations can\n",
       "be found in any statistics text.</p>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.random\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">random</code><span class=\"sig-paren\">(</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.random\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return the next random floating point number in the range [0.0, 1.0).</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.uniform\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">uniform</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a</em>, <em class=\"sig-param\">b</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.uniform\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return a random floating point number <em>N</em> such that <code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">b</span></code> for\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">b</span></code> and <code class=\"docutils literal notranslate\"><span class=\"pre\">b</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">a</span></code> for <code class=\"docutils literal notranslate\"><span class=\"pre\">b</span> <span class=\"pre\">&lt;</span> <span class=\"pre\">a</span></code>.</p>\n",
       "<p>The end-point value <code class=\"docutils literal notranslate\"><span class=\"pre\">b</span></code> may or may not be included in the range\n",
       "depending on floating-point rounding in the equation <code class=\"docutils literal notranslate\"><span class=\"pre\">a</span> <span class=\"pre\">+</span> <span class=\"pre\">(b-a)</span> <span class=\"pre\">*</span> <span class=\"pre\">random()</span></code>.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.triangular\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">triangular</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">low</em>, <em class=\"sig-param\">high</em>, <em class=\"sig-param\">mode</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.triangular\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Return a random floating point number <em>N</em> such that <code class=\"docutils literal notranslate\"><span class=\"pre\">low</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">N</span> <span class=\"pre\">&lt;=</span> <span class=\"pre\">high</span></code> and\n",
       "with the specified <em>mode</em> between those bounds.  The <em>low</em> and <em>high</em> bounds\n",
       "default to zero and one.  The <em>mode</em> argument defaults to the midpoint\n",
       "between the bounds, giving a symmetric distribution.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.betavariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">betavariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.betavariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Beta distribution.  Conditions on the parameters are <code class=\"docutils literal notranslate\"><span class=\"pre\">alpha</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code> and\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">beta</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code>. Returned values range between 0 and 1.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.expovariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">expovariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">lambd</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.expovariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Exponential distribution.  <em>lambd</em> is 1.0 divided by the desired\n",
       "mean.  It should be nonzero.  (The parameter would be called\n",
       "“lambda”, but that is a reserved word in Python.)  Returned values\n",
       "range from 0 to positive infinity if <em>lambd</em> is positive, and from\n",
       "negative infinity to 0 if <em>lambd</em> is negative.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.gammavariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">gammavariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.gammavariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Gamma distribution.  (<em>Not</em> the gamma function!)  Conditions on the\n",
       "parameters are <code class=\"docutils literal notranslate\"><span class=\"pre\">alpha</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code> and <code class=\"docutils literal notranslate\"><span class=\"pre\">beta</span> <span class=\"pre\">&gt;</span> <span class=\"pre\">0</span></code>.</p>\n",
       "<p>The probability distribution function is:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span>          <span class=\"n\">x</span> <span class=\"o\">**</span> <span class=\"p\">(</span><span class=\"n\">alpha</span> <span class=\"o\">-</span> <span class=\"mi\">1</span><span class=\"p\">)</span> <span class=\"o\">*</span> <span class=\"n\">math</span><span class=\"o\">.</span><span class=\"n\">exp</span><span class=\"p\">(</span><span class=\"o\">-</span><span class=\"n\">x</span> <span class=\"o\">/</span> <span class=\"n\">beta</span><span class=\"p\">)</span>\n",
       "<span class=\"n\">pdf</span><span class=\"p\">(</span><span class=\"n\">x</span><span class=\"p\">)</span> <span class=\"o\">=</span>  <span class=\"o\">--------------------------------------</span>\n",
       "            <span class=\"n\">math</span><span class=\"o\">.</span><span class=\"n\">gamma</span><span class=\"p\">(</span><span class=\"n\">alpha</span><span class=\"p\">)</span> <span class=\"o\">*</span> <span class=\"n\">beta</span> <span class=\"o\">**</span> <span class=\"n\">alpha</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.gauss\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">gauss</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.gauss\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Gaussian distribution.  <em>mu</em> is the mean, and <em>sigma</em> is the standard\n",
       "deviation.  This is slightly faster than the <a class=\"reference internal\" href=\"#random.normalvariate\" title=\"random.normalvariate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">normalvariate()</span></code></a> function\n",
       "defined below.</p>\n",
       "<p>Multithreading note:  When two threads call this function\n",
       "simultaneously, it is possible that they will receive the\n",
       "same return value.  This can be avoided in three ways.\n",
       "1) Have each thread use a different instance of the random\n",
       "number generator. 2) Put locks around all calls. 3) Use the\n",
       "slower, but thread-safe <a class=\"reference internal\" href=\"#random.normalvariate\" title=\"random.normalvariate\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">normalvariate()</span></code></a> function instead.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.lognormvariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">lognormvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.lognormvariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Log normal distribution.  If you take the natural logarithm of this\n",
       "distribution, you’ll get a normal distribution with mean <em>mu</em> and standard\n",
       "deviation <em>sigma</em>.  <em>mu</em> can have any value, and <em>sigma</em> must be greater than\n",
       "zero.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.normalvariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">normalvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.normalvariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Normal distribution.  <em>mu</em> is the mean, and <em>sigma</em> is the standard deviation.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.vonmisesvariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">vonmisesvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">kappa</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.vonmisesvariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p><em>mu</em> is the mean angle, expressed in radians between 0 and 2*<em>pi</em>, and <em>kappa</em>\n",
       "is the concentration parameter, which must be greater than or equal to zero.  If\n",
       "<em>kappa</em> is equal to zero, this distribution reduces to a uniform random angle\n",
       "over the range 0 to 2*<em>pi</em>.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.paretovariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">paretovariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.paretovariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Pareto distribution.  <em>alpha</em> is the shape parameter.</p>\n",
       "</dd></dl>\n",
       "<dl class=\"function\">\n",
       "<dt id=\"random.weibullvariate\">\n",
       "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">weibullvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.weibullvariate\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Weibull distribution.  <em>alpha</em> is the scale parameter and <em>beta</em> is the shape\n",
       "parameter.</p>\n",
       "</dd></dl>\n",
       "</div>\n",
       "<div class=\"section\" id=\"alternative-generator\">\n",
       "<h2>Alternative Generator<a class=\"headerlink\" href=\"#alternative-generator\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<dl class=\"class\">\n",
       "<dt id=\"random.Random\">\n",
       "<em class=\"property\">class </em><code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">Random</code><span class=\"sig-paren\">(</span><span class=\"optional\">[</span><em class=\"sig-param\">seed</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.Random\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Class that implements the default pseudo-random number generator used by the\n",
       "<a class=\"reference internal\" href=\"#module-random\" title=\"random: Generate pseudo-random numbers with various common distributions.\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code></a> module.</p>\n",
       "<div class=\"deprecated\">\n",
       "<p><span class=\"versionmodified deprecated\">Deprecated since version 3.9: </span>In the future, the <em>seed</em> must be one of the following types:\n",
       "<code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">NoneType</span></code>, <a class=\"reference internal\" href=\"functions.html#int\" title=\"int\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">int</span></code></a>, <a class=\"reference internal\" href=\"functions.html#float\" title=\"float\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">float</span></code></a>, <a class=\"reference internal\" href=\"stdtypes.html#str\" title=\"str\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">str</span></code></a>,\n",
       "<a class=\"reference internal\" href=\"stdtypes.html#bytes\" title=\"bytes\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytes</span></code></a>, or <a class=\"reference internal\" href=\"stdtypes.html#bytearray\" title=\"bytearray\"><code class=\"xref py py-class docutils literal notranslate\"><span class=\"pre\">bytearray</span></code></a>.</p>\n",
       "</div>\n",
       "</dd></dl>\n",
       "<dl class=\"class\">\n",
       "<dt id=\"random.SystemRandom\">\n",
       "<em class=\"property\">class </em><code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">SystemRandom</code><span class=\"sig-paren\">(</span><span class=\"optional\">[</span><em class=\"sig-param\">seed</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.SystemRandom\" title=\"Permalink to this definition\">¶</a></dt>\n",
       "<dd><p>Class that uses the <a class=\"reference internal\" href=\"os.html#os.urandom\" title=\"os.urandom\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">os.urandom()</span></code></a> function for generating random numbers\n",
       "from sources provided by the operating system. Not available on all systems.\n",
       "Does not rely on software state, and sequences are not reproducible. Accordingly,\n",
       "the <a class=\"reference internal\" href=\"#random.seed\" title=\"random.seed\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">seed()</span></code></a> method has no effect and is ignored.\n",
       "The <a class=\"reference internal\" href=\"#random.getstate\" title=\"random.getstate\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">getstate()</span></code></a> and <a class=\"reference internal\" href=\"#random.setstate\" title=\"random.setstate\"><code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">setstate()</span></code></a> methods raise\n",
       "<a class=\"reference internal\" href=\"exceptions.html#NotImplementedError\" title=\"NotImplementedError\"><code class=\"xref py py-exc docutils literal notranslate\"><span class=\"pre\">NotImplementedError</span></code></a> if called.</p>\n",
       "</dd></dl>\n",
       "</div>\n",
       "<div class=\"section\" id=\"notes-on-reproducibility\">\n",
       "<h2>Notes on Reproducibility<a class=\"headerlink\" href=\"#notes-on-reproducibility\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<p>Sometimes it is useful to be able to reproduce the sequences given by a\n",
       "pseudo-random number generator.  By re-using a seed value, the same sequence should be\n",
       "reproducible from run to run as long as multiple threads are not running.</p>\n",
       "<p>Most of the random module’s algorithms and seeding functions are subject to\n",
       "change across Python versions, but two aspects are guaranteed not to change:</p>\n",
       "<ul class=\"simple\">\n",
       "<li><p>If a new seeding method is added, then a backward compatible seeder will be\n",
       "offered.</p></li>\n",
       "<li><p>The generator’s <code class=\"xref py py-meth docutils literal notranslate\"><span class=\"pre\">random()</span></code> method will continue to produce the same\n",
       "sequence when the compatible seeder is given the same seed.</p></li>\n",
       "</ul>\n",
       "</div>\n",
       "<div class=\"section\" id=\"examples\">\n",
       "<span id=\"random-examples\"></span><h2>Examples<a class=\"headerlink\" href=\"#examples\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<p>Basic examples:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">random</span><span class=\"p\">()</span>                             <span class=\"c1\"># Random float:  0.0 &lt;= x &lt; 1.0</span>\n",
       "<span class=\"go\">0.37444887175646646</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">uniform</span><span class=\"p\">(</span><span class=\"mf\">2.5</span><span class=\"p\">,</span> <span class=\"mf\">10.0</span><span class=\"p\">)</span>                   <span class=\"c1\"># Random float:  2.5 &lt;= x &lt; 10.0</span>\n",
       "<span class=\"go\">3.1800146073117523</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">expovariate</span><span class=\"p\">(</span><span class=\"mi\">1</span> <span class=\"o\">/</span> <span class=\"mi\">5</span><span class=\"p\">)</span>                   <span class=\"c1\"># Interval between arrivals averaging 5 seconds</span>\n",
       "<span class=\"go\">5.148957571865031</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">10</span><span class=\"p\">)</span>                        <span class=\"c1\"># Integer from 0 to 9 inclusive</span>\n",
       "<span class=\"go\">7</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">randrange</span><span class=\"p\">(</span><span class=\"mi\">0</span><span class=\"p\">,</span> <span class=\"mi\">101</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">)</span>                 <span class=\"c1\"># Even integer from 0 to 100 inclusive</span>\n",
       "<span class=\"go\">26</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">choice</span><span class=\"p\">([</span><span class=\"s1\">'win'</span><span class=\"p\">,</span> <span class=\"s1\">'lose'</span><span class=\"p\">,</span> <span class=\"s1\">'draw'</span><span class=\"p\">])</span>      <span class=\"c1\"># Single random element from a sequence</span>\n",
       "<span class=\"go\">'draw'</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">deck</span> <span class=\"o\">=</span> <span class=\"s1\">'ace two three four'</span><span class=\"o\">.</span><span class=\"n\">split</span><span class=\"p\">()</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">shuffle</span><span class=\"p\">(</span><span class=\"n\">deck</span><span class=\"p\">)</span>                        <span class=\"c1\"># Shuffle a list</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">deck</span>\n",
       "<span class=\"go\">['four', 'two', 'ace', 'three']</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">sample</span><span class=\"p\">([</span><span class=\"mi\">10</span><span class=\"p\">,</span> <span class=\"mi\">20</span><span class=\"p\">,</span> <span class=\"mi\">30</span><span class=\"p\">,</span> <span class=\"mi\">40</span><span class=\"p\">,</span> <span class=\"mi\">50</span><span class=\"p\">],</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">4</span><span class=\"p\">)</span>    <span class=\"c1\"># Four samples without replacement</span>\n",
       "<span class=\"go\">[40, 10, 50, 30]</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<p>Simulations:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Six roulette wheel spins (weighted sampling with replacement)</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">choices</span><span class=\"p\">([</span><span class=\"s1\">'red'</span><span class=\"p\">,</span> <span class=\"s1\">'black'</span><span class=\"p\">,</span> <span class=\"s1\">'green'</span><span class=\"p\">],</span> <span class=\"p\">[</span><span class=\"mi\">18</span><span class=\"p\">,</span> <span class=\"mi\">18</span><span class=\"p\">,</span> <span class=\"mi\">2</span><span class=\"p\">],</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">6</span><span class=\"p\">)</span>\n",
       "<span class=\"go\">['red', 'green', 'black', 'black', 'red', 'black']</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Deal 20 cards without replacement from a deck</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># of 52 playing cards, and determine the proportion of cards</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># with a ten-value:  ten, jack, queen, or king.</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">dealt</span> <span class=\"o\">=</span> <span class=\"n\">sample</span><span class=\"p\">([</span><span class=\"s1\">'tens'</span><span class=\"p\">,</span> <span class=\"s1\">'low cards'</span><span class=\"p\">],</span> <span class=\"n\">counts</span><span class=\"o\">=</span><span class=\"p\">[</span><span class=\"mi\">16</span><span class=\"p\">,</span> <span class=\"mi\">36</span><span class=\"p\">],</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">20</span><span class=\"p\">)</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">dealt</span><span class=\"o\">.</span><span class=\"n\">count</span><span class=\"p\">(</span><span class=\"s1\">'tens'</span><span class=\"p\">)</span> <span class=\"o\">/</span> <span class=\"mi\">20</span>\n",
       "<span class=\"go\">0.15</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Estimate the probability of getting 5 or more heads from 7 spins</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># of a biased coin that settles on heads 60% of the time.</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">trial</span><span class=\"p\">():</span>\n",
       "<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"n\">choices</span><span class=\"p\">(</span><span class=\"s1\">'HT'</span><span class=\"p\">,</span> <span class=\"n\">cum_weights</span><span class=\"o\">=</span><span class=\"p\">(</span><span class=\"mf\">0.60</span><span class=\"p\">,</span> <span class=\"mf\">1.00</span><span class=\"p\">),</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">7</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">count</span><span class=\"p\">(</span><span class=\"s1\">'H'</span><span class=\"p\">)</span> <span class=\"o\">&gt;=</span> <span class=\"mi\">5</span>\n",
       "<span class=\"gp\">...</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">sum</span><span class=\"p\">(</span><span class=\"n\">trial</span><span class=\"p\">()</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10_000</span><span class=\"p\">))</span> <span class=\"o\">/</span> <span class=\"mi\">10_000</span>\n",
       "<span class=\"go\">0.4169</span>\n",
       "\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"c1\"># Probability of the median of 5 samples being in middle two quartiles</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"k\">def</span> <span class=\"nf\">trial</span><span class=\"p\">():</span>\n",
       "<span class=\"gp\">... </span>    <span class=\"k\">return</span> <span class=\"mi\">2_500</span> <span class=\"o\">&lt;=</span> <span class=\"nb\">sorted</span><span class=\"p\">(</span><span class=\"n\">choices</span><span class=\"p\">(</span><span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10_000</span><span class=\"p\">),</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"mi\">5</span><span class=\"p\">))[</span><span class=\"mi\">2</span><span class=\"p\">]</span> <span class=\"o\">&lt;</span> <span class=\"mi\">7_500</span>\n",
       "<span class=\"gp\">...</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"nb\">sum</span><span class=\"p\">(</span><span class=\"n\">trial</span><span class=\"p\">()</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">10_000</span><span class=\"p\">))</span> <span class=\"o\">/</span> <span class=\"mi\">10_000</span>\n",
       "<span class=\"go\">0.7958</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<p>Example of <a class=\"reference external\" href=\"https://en.wikipedia.org/wiki/Bootstrapping_(statistics)\">statistical bootstrapping</a> using resampling\n",
       "with replacement to estimate a confidence interval for the mean of a sample:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"c1\"># http://statistics.about.com/od/Applications/a/Example-Of-Bootstrapping.htm</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">statistics</span> <span class=\"kn\">import</span> <span class=\"n\">fmean</span> <span class=\"k\">as</span> <span class=\"n\">mean</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">choices</span>\n",
       "\n",
       "<span class=\"n\">data</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">41</span><span class=\"p\">,</span> <span class=\"mi\">50</span><span class=\"p\">,</span> <span class=\"mi\">29</span><span class=\"p\">,</span> <span class=\"mi\">37</span><span class=\"p\">,</span> <span class=\"mi\">81</span><span class=\"p\">,</span> <span class=\"mi\">30</span><span class=\"p\">,</span> <span class=\"mi\">73</span><span class=\"p\">,</span> <span class=\"mi\">63</span><span class=\"p\">,</span> <span class=\"mi\">20</span><span class=\"p\">,</span> <span class=\"mi\">35</span><span class=\"p\">,</span> <span class=\"mi\">68</span><span class=\"p\">,</span> <span class=\"mi\">22</span><span class=\"p\">,</span> <span class=\"mi\">60</span><span class=\"p\">,</span> <span class=\"mi\">31</span><span class=\"p\">,</span> <span class=\"mi\">95</span><span class=\"p\">]</span>\n",
       "<span class=\"n\">means</span> <span class=\"o\">=</span> <span class=\"nb\">sorted</span><span class=\"p\">(</span><span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">choices</span><span class=\"p\">(</span><span class=\"n\">data</span><span class=\"p\">,</span> <span class=\"n\">k</span><span class=\"o\">=</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">data</span><span class=\"p\">)))</span> <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">100</span><span class=\"p\">))</span>\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'The sample mean of </span><span class=\"si\">{</span><span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">data</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\"> has a 90% confidence '</span>\n",
       "      <span class=\"sa\">f</span><span class=\"s1\">'interval from </span><span class=\"si\">{</span><span class=\"n\">means</span><span class=\"p\">[</span><span class=\"mi\">5</span><span class=\"p\">]</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\"> to </span><span class=\"si\">{</span><span class=\"n\">means</span><span class=\"p\">[</span><span class=\"mi\">94</span><span class=\"p\">]</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">'</span><span class=\"p\">)</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<p>Example of a <a class=\"reference external\" href=\"https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests\">resampling permutation test</a>\n",
       "to determine the statistical significance or <a class=\"reference external\" href=\"https://en.wikipedia.org/wiki/P-value\">p-value</a> of an observed difference\n",
       "between the effects of a drug versus a placebo:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"c1\"># Example from \"Statistics is Easy\" by Dennis Shasha and Manda Wilson</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">statistics</span> <span class=\"kn\">import</span> <span class=\"n\">fmean</span> <span class=\"k\">as</span> <span class=\"n\">mean</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">shuffle</span>\n",
       "\n",
       "<span class=\"n\">drug</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">54</span><span class=\"p\">,</span> <span class=\"mi\">73</span><span class=\"p\">,</span> <span class=\"mi\">53</span><span class=\"p\">,</span> <span class=\"mi\">70</span><span class=\"p\">,</span> <span class=\"mi\">73</span><span class=\"p\">,</span> <span class=\"mi\">68</span><span class=\"p\">,</span> <span class=\"mi\">52</span><span class=\"p\">,</span> <span class=\"mi\">65</span><span class=\"p\">,</span> <span class=\"mi\">65</span><span class=\"p\">]</span>\n",
       "<span class=\"n\">placebo</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mi\">54</span><span class=\"p\">,</span> <span class=\"mi\">51</span><span class=\"p\">,</span> <span class=\"mi\">58</span><span class=\"p\">,</span> <span class=\"mi\">44</span><span class=\"p\">,</span> <span class=\"mi\">55</span><span class=\"p\">,</span> <span class=\"mi\">52</span><span class=\"p\">,</span> <span class=\"mi\">42</span><span class=\"p\">,</span> <span class=\"mi\">47</span><span class=\"p\">,</span> <span class=\"mi\">58</span><span class=\"p\">,</span> <span class=\"mi\">46</span><span class=\"p\">]</span>\n",
       "<span class=\"n\">observed_diff</span> <span class=\"o\">=</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">drug</span><span class=\"p\">)</span> <span class=\"o\">-</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">placebo</span><span class=\"p\">)</span>\n",
       "\n",
       "<span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"mi\">10_000</span>\n",
       "<span class=\"n\">count</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\n",
       "<span class=\"n\">combined</span> <span class=\"o\">=</span> <span class=\"n\">drug</span> <span class=\"o\">+</span> <span class=\"n\">placebo</span>\n",
       "<span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">):</span>\n",
       "    <span class=\"n\">shuffle</span><span class=\"p\">(</span><span class=\"n\">combined</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">new_diff</span> <span class=\"o\">=</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">combined</span><span class=\"p\">[:</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">drug</span><span class=\"p\">)])</span> <span class=\"o\">-</span> <span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">combined</span><span class=\"p\">[</span><span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">drug</span><span class=\"p\">):])</span>\n",
       "    <span class=\"n\">count</span> <span class=\"o\">+=</span> <span class=\"p\">(</span><span class=\"n\">new_diff</span> <span class=\"o\">&gt;=</span> <span class=\"n\">observed_diff</span><span class=\"p\">)</span>\n",
       "\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'</span><span class=\"si\">{</span><span class=\"n\">n</span><span class=\"si\">}</span><span class=\"s1\"> label reshufflings produced only </span><span class=\"si\">{</span><span class=\"n\">count</span><span class=\"si\">}</span><span class=\"s1\"> instances with a difference'</span><span class=\"p\">)</span>\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'at least as extreme as the observed difference of </span><span class=\"si\">{</span><span class=\"n\">observed_diff</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.'</span><span class=\"p\">)</span>\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'The one-sided p-value of </span><span class=\"si\">{</span><span class=\"n\">count</span> <span class=\"o\">/</span> <span class=\"n\">n</span><span class=\"si\">:</span><span class=\"s1\">.4f</span><span class=\"si\">}</span><span class=\"s1\"> leads us to reject the null'</span><span class=\"p\">)</span>\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'hypothesis that there is no difference between the drug and the placebo.'</span><span class=\"p\">)</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<p>Simulation of arrival times and service deliveries for a multiserver queue:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"kn\">from</span> <span class=\"nn\">heapq</span> <span class=\"kn\">import</span> <span class=\"n\">heappush</span><span class=\"p\">,</span> <span class=\"n\">heappop</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">expovariate</span><span class=\"p\">,</span> <span class=\"n\">gauss</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">statistics</span> <span class=\"kn\">import</span> <span class=\"n\">mean</span><span class=\"p\">,</span> <span class=\"n\">median</span><span class=\"p\">,</span> <span class=\"n\">stdev</span>\n",
       "\n",
       "<span class=\"n\">average_arrival_interval</span> <span class=\"o\">=</span> <span class=\"mf\">5.6</span>\n",
       "<span class=\"n\">average_service_time</span> <span class=\"o\">=</span> <span class=\"mf\">15.0</span>\n",
       "<span class=\"n\">stdev_service_time</span> <span class=\"o\">=</span> <span class=\"mf\">3.5</span>\n",
       "<span class=\"n\">num_servers</span> <span class=\"o\">=</span> <span class=\"mi\">3</span>\n",
       "\n",
       "<span class=\"n\">waits</span> <span class=\"o\">=</span> <span class=\"p\">[]</span>\n",
       "<span class=\"n\">arrival_time</span> <span class=\"o\">=</span> <span class=\"mf\">0.0</span>\n",
       "<span class=\"n\">servers</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"mf\">0.0</span><span class=\"p\">]</span> <span class=\"o\">*</span> <span class=\"n\">num_servers</span>  <span class=\"c1\"># time when each server becomes available</span>\n",
       "<span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"mi\">100_000</span><span class=\"p\">):</span>\n",
       "    <span class=\"n\">arrival_time</span> <span class=\"o\">+=</span> <span class=\"n\">expovariate</span><span class=\"p\">(</span><span class=\"mf\">1.0</span> <span class=\"o\">/</span> <span class=\"n\">average_arrival_interval</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">next_server_available</span> <span class=\"o\">=</span> <span class=\"n\">heappop</span><span class=\"p\">(</span><span class=\"n\">servers</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">wait</span> <span class=\"o\">=</span> <span class=\"nb\">max</span><span class=\"p\">(</span><span class=\"mf\">0.0</span><span class=\"p\">,</span> <span class=\"n\">next_server_available</span> <span class=\"o\">-</span> <span class=\"n\">arrival_time</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">waits</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">wait</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">service_duration</span> <span class=\"o\">=</span> <span class=\"n\">gauss</span><span class=\"p\">(</span><span class=\"n\">average_service_time</span><span class=\"p\">,</span> <span class=\"n\">stdev_service_time</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">service_completed</span> <span class=\"o\">=</span> <span class=\"n\">arrival_time</span> <span class=\"o\">+</span> <span class=\"n\">wait</span> <span class=\"o\">+</span> <span class=\"n\">service_duration</span>\n",
       "    <span class=\"n\">heappush</span><span class=\"p\">(</span><span class=\"n\">servers</span><span class=\"p\">,</span> <span class=\"n\">service_completed</span><span class=\"p\">)</span>\n",
       "\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'Mean wait: </span><span class=\"si\">{</span><span class=\"n\">mean</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.  Stdev wait: </span><span class=\"si\">{</span><span class=\"n\">stdev</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.'</span><span class=\"p\">)</span>\n",
       "<span class=\"nb\">print</span><span class=\"p\">(</span><span class=\"sa\">f</span><span class=\"s1\">'Median wait: </span><span class=\"si\">{</span><span class=\"n\">median</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.  Max wait: </span><span class=\"si\">{</span><span class=\"nb\">max</span><span class=\"p\">(</span><span class=\"n\">waits</span><span class=\"p\">)</span><span class=\"si\">:</span><span class=\"s1\">.1f</span><span class=\"si\">}</span><span class=\"s1\">.'</span><span class=\"p\">)</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<div class=\"admonition seealso\">\n",
       "<p class=\"admonition-title\">See also</p>\n",
       "<p><a class=\"reference external\" href=\"https://www.youtube.com/watch?v=Iq9DzN6mvYA\">Statistics for Hackers</a>\n",
       "a video tutorial by\n",
       "<a class=\"reference external\" href=\"https://us.pycon.org/2016/speaker/profile/295/\">Jake Vanderplas</a>\n",
       "on statistical analysis using just a few fundamental concepts\n",
       "including simulation, sampling, shuffling, and cross-validation.</p>\n",
       "<p><a class=\"reference external\" href=\"http://nbviewer.jupyter.org/url/norvig.com/ipython/Economics.ipynb\">Economics Simulation</a>\n",
       "a simulation of a marketplace by\n",
       "<a class=\"reference external\" href=\"http://norvig.com/bio.html\">Peter Norvig</a> that shows effective\n",
       "use of many of the tools and distributions provided by this module\n",
       "(gauss, uniform, sample, betavariate, choice, triangular, and randrange).</p>\n",
       "<p><a class=\"reference external\" href=\"http://nbviewer.jupyter.org/url/norvig.com/ipython/Probability.ipynb\">A Concrete Introduction to Probability (using Python)</a>\n",
       "a tutorial by <a class=\"reference external\" href=\"http://norvig.com/bio.html\">Peter Norvig</a> covering\n",
       "the basics of probability theory, how to write simulations, and\n",
       "how to perform data analysis using Python.</p>\n",
       "</div>\n",
       "</div>\n",
       "<div class=\"section\" id=\"recipes\">\n",
       "<h2>Recipes<a class=\"headerlink\" href=\"#recipes\" title=\"Permalink to this headline\">¶</a></h2>\n",
       "<p>The default <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a> returns multiples of 2⁻⁵³ in the range\n",
       "<em>0.0 ≤ x &lt; 1.0</em>.  All such numbers are evenly spaced and are exactly\n",
       "representable as Python floats.  However, many other representable\n",
       "floats in that interval are not possible selections.  For example,\n",
       "<code class=\"docutils literal notranslate\"><span class=\"pre\">0.05954861408025609</span></code> isn’t an integer multiple of 2⁻⁵³.</p>\n",
       "<p>The following recipe takes a different approach.  All floats in the\n",
       "interval are possible selections.  The mantissa comes from a uniform\n",
       "distribution of integers in the range <em>2⁵² ≤ mantissa &lt; 2⁵³</em>.  The\n",
       "exponent comes from a geometric distribution where exponents smaller\n",
       "than <em>-53</em> occur half as often as the next larger exponent.</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"kn\">from</span> <span class=\"nn\">random</span> <span class=\"kn\">import</span> <span class=\"n\">Random</span>\n",
       "<span class=\"kn\">from</span> <span class=\"nn\">math</span> <span class=\"kn\">import</span> <span class=\"n\">ldexp</span>\n",
       "\n",
       "<span class=\"k\">class</span> <span class=\"nc\">FullRandom</span><span class=\"p\">(</span><span class=\"n\">Random</span><span class=\"p\">):</span>\n",
       "\n",
       "    <span class=\"k\">def</span> <span class=\"nf\">random</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">):</span>\n",
       "        <span class=\"n\">mantissa</span> <span class=\"o\">=</span> <span class=\"mh\">0x10_0000_0000_0000</span> <span class=\"o\">|</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">getrandbits</span><span class=\"p\">(</span><span class=\"mi\">52</span><span class=\"p\">)</span>\n",
       "        <span class=\"n\">exponent</span> <span class=\"o\">=</span> <span class=\"o\">-</span><span class=\"mi\">53</span>\n",
       "        <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"mi\">0</span>\n",
       "        <span class=\"k\">while</span> <span class=\"ow\">not</span> <span class=\"n\">x</span><span class=\"p\">:</span>\n",
       "            <span class=\"n\">x</span> <span class=\"o\">=</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">getrandbits</span><span class=\"p\">(</span><span class=\"mi\">32</span><span class=\"p\">)</span>\n",
       "            <span class=\"n\">exponent</span> <span class=\"o\">+=</span> <span class=\"n\">x</span><span class=\"o\">.</span><span class=\"n\">bit_length</span><span class=\"p\">()</span> <span class=\"o\">-</span> <span class=\"mi\">32</span>\n",
       "        <span class=\"k\">return</span> <span class=\"n\">ldexp</span><span class=\"p\">(</span><span class=\"n\">mantissa</span><span class=\"p\">,</span> <span class=\"n\">exponent</span><span class=\"p\">)</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<p>All <a class=\"reference internal\" href=\"#real-valued-distributions\"><span class=\"std std-ref\">real valued distributions</span></a>\n",
       "in the class will use the new method:</p>\n",
       "<div class=\"highlight-python3 notranslate\"><div class=\"highlight\"><pre><span></span><span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fr</span> <span class=\"o\">=</span> <span class=\"n\">FullRandom</span><span class=\"p\">()</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fr</span><span class=\"o\">.</span><span class=\"n\">random</span><span class=\"p\">()</span>\n",
       "<span class=\"go\">0.05954861408025609</span>\n",
       "<span class=\"gp\">&gt;&gt;&gt; </span><span class=\"n\">fr</span><span class=\"o\">.</span><span class=\"n\">expovariate</span><span class=\"p\">(</span><span class=\"mf\">0.25</span><span class=\"p\">)</span>\n",
       "<span class=\"go\">8.87925541791544</span>\n",
       "</pre></div>\n",
       "</div>\n",
       "<p>The recipe is conceptually equivalent to an algorithm that chooses from\n",
       "all the multiples of 2⁻¹⁰⁷⁴ in the range <em>0.0 ≤ x &lt; 1.0</em>.  All such\n",
       "numbers are evenly spaced, but most have to be rounded down to the\n",
       "nearest representable Python float.  (The value 2⁻¹⁰⁷⁴ is the smallest\n",
       "positive unnormalized float and is equal to <code class=\"docutils literal notranslate\"><span class=\"pre\">math.ulp(0.0)</span></code>.)</p>\n",
       "<div class=\"admonition seealso\">\n",
       "<p class=\"admonition-title\">See also</p>\n",
       "<p><a class=\"reference external\" href=\"https://allendowney.com/research/rand/downey07randfloat.pdf\">Generating Pseudo-random Floating-Point Values</a> a\n",
       "paper by Allen B. Downey describing ways to generate more\n",
       "fine-grained floats than normally generated by <a class=\"reference internal\" href=\"#random.random\" title=\"random.random\"><code class=\"xref py py-func docutils literal notranslate\"><span class=\"pre\">random()</span></code></a>.</p>\n",
       "</div>\n",
       "</div>\n",
       "</div>\n",
       "</div>\n",
       "</div>\n",
       "</div>\n",
       "<div aria-label=\"main navigation\" class=\"sphinxsidebar\" role=\"navigation\">\n",
       "<div class=\"sphinxsidebarwrapper\">\n",
       "<h3><a href=\"../contents.html\">Table of Contents</a></h3>\n",
       "<ul>\n",
       "<li><a class=\"reference internal\" href=\"#\"><code class=\"xref py py-mod docutils literal notranslate\"><span class=\"pre\">random</span></code> — Generate pseudo-random numbers</a><ul>\n",
       "<li><a class=\"reference internal\" href=\"#bookkeeping-functions\">Bookkeeping functions</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#functions-for-bytes\">Functions for bytes</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#functions-for-integers\">Functions for integers</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#functions-for-sequences\">Functions for sequences</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#real-valued-distributions\">Real-valued distributions</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#alternative-generator\">Alternative Generator</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#notes-on-reproducibility\">Notes on Reproducibility</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#examples\">Examples</a></li>\n",
       "<li><a class=\"reference internal\" href=\"#recipes\">Recipes</a></li>\n",
       "</ul>\n",
       "</li>\n",
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check soup variable\n",
    "\n",
    "soup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Other way to load the html code using 'urllib.request.urlopen()'\n",
    "\n",
    "#url = urllib.request.urlopen(\"https://docs.python.org/3/library/random.html#module-random\")\n",
    "#soup = BeautifulSoup(url)\n",
    "#soup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Still very long, but a little easier to take in.**\n",
    "\n",
    "The real advantage of Beautiful Soup, however, is that it *parses* our webpage according to its structure and allows us to *search for* and *extract* elements within it. This is because it transforms the webpage from a string into a special Beautiful Soup object."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To extract HTML elements from our webpage, we can call the `.find()` method on our Beautiful Soup object. This method finds the first element that matches the criterion that we pass in. The criterion may be an element `id`, `class`, tag `name`, or even a function. (For a full list of search elements, see [this page](https://www.crummy.com/software/BeautifulSoup/bs4/doc/#searching-the-tree).)\n",
    "\n",
    "But how do we know what element to search for? This is where your browser's `Inspect` or `Inspect Element` feature comes in handy. Simply right click on an object of interest on the web page and click `Inspect` on Chrome or `Inspect Element` on Firefox. This will then show you the corresponding place in the HTML code where the element appears. From there you should be able to find an id or class name that will allow you to locate the element with Beautiful Soup.\n",
    "\n",
    "**In this case, we want to target the tag/ element `dt` as below picture:**\n",
    "\n",
    "<br>\n",
    "<br>\n",
    "<img src = \"https://docs.google.com/uc?export=download&id=1Mj9K4QnhS5mFMK4Ddx_6OHFUIN-xpR9X\" />\n",
    "\n",
    "<br>\n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**So it looks like we're looking for a `dt` element with `id='random.___'`. We can easily retrieve this with Beautiful Soup's `.findAll` command.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    },
    "scrolled": true,
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[<dt id=\"random.seed\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">seed</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a=None</em>, <em class=\"sig-param\">version=2</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.seed\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.getstate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">getstate</code><span class=\"sig-paren\">(</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.getstate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.setstate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">setstate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">state</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.setstate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.randbytes\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randbytes</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">n</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randbytes\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.randrange\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randrange</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">stop</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randrange\" title=\"Permalink to this definition\">¶</a></dt>, <dt>\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randrange</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">start</em>, <em class=\"sig-param\">stop</em><span class=\"optional\">[</span>, <em class=\"sig-param\">step</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span></dt>, <dt id=\"random.randint\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">randint</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a</em>, <em class=\"sig-param\">b</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.randint\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.getrandbits\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">getrandbits</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">k</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.getrandbits\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.choice\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">choice</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">seq</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.choice\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.choices\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">choices</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">population</em>, <em class=\"sig-param\">weights=None</em>, <em class=\"sig-param\">*</em>, <em class=\"sig-param\">cum_weights=None</em>, <em class=\"sig-param\">k=1</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.choices\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.shuffle\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">shuffle</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">x</em><span class=\"optional\">[</span>, <em class=\"sig-param\">random</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.shuffle\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.sample\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">sample</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">population</em>, <em class=\"sig-param\">k</em>, <em class=\"sig-param\">*</em>, <em class=\"sig-param\">counts=None</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.sample\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.random\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">random</code><span class=\"sig-paren\">(</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.random\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.uniform\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">uniform</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">a</em>, <em class=\"sig-param\">b</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.uniform\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.triangular\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">triangular</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">low</em>, <em class=\"sig-param\">high</em>, <em class=\"sig-param\">mode</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.triangular\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.betavariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">betavariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.betavariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.expovariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">expovariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">lambd</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.expovariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.gammavariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">gammavariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.gammavariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.gauss\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">gauss</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.gauss\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.lognormvariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">lognormvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.lognormvariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.normalvariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">normalvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">sigma</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.normalvariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.vonmisesvariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">vonmisesvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">mu</em>, <em class=\"sig-param\">kappa</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.vonmisesvariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.paretovariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">paretovariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.paretovariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.weibullvariate\">\n",
      "<code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">weibullvariate</code><span class=\"sig-paren\">(</span><em class=\"sig-param\">alpha</em>, <em class=\"sig-param\">beta</em><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.weibullvariate\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.Random\">\n",
      "<em class=\"property\">class </em><code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">Random</code><span class=\"sig-paren\">(</span><span class=\"optional\">[</span><em class=\"sig-param\">seed</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.Random\" title=\"Permalink to this definition\">¶</a></dt>, <dt id=\"random.SystemRandom\">\n",
      "<em class=\"property\">class </em><code class=\"sig-prename descclassname\">random.</code><code class=\"sig-name descname\">SystemRandom</code><span class=\"sig-paren\">(</span><span class=\"optional\">[</span><em class=\"sig-param\">seed</em><span class=\"optional\">]</span><span class=\"sig-paren\">)</span><a class=\"headerlink\" href=\"#random.SystemRandom\" title=\"Permalink to this definition\">¶</a></dt>]\n"
     ]
    }
   ],
   "source": [
    "# Find all function names - we specify the name of the element in this case is 'dt'\n",
    "\n",
    "names = soup.body.findAll('dt')\n",
    "\n",
    "print(names)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**There are still some works to do! This is when regex kicks in.**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['id=\"random.seed', 'id=\"random.getstate', 'id=\"random.setstate', 'id=\"random.randbytes', 'id=\"random.randrange', 'id=\"random.randint', 'id=\"random.getrandbits', 'id=\"random.choice', 'id=\"random.choices', 'id=\"random.shuffle', 'id=\"random.sample', 'id=\"random.random', 'id=\"random.uniform', 'id=\"random.triangular', 'id=\"random.betavariate', 'id=\"random.expovariate', 'id=\"random.gammavariate', 'id=\"random.gauss', 'id=\"random.lognormvariate', 'id=\"random.normalvariate', 'id=\"random.vonmisesvariate', 'id=\"random.paretovariate', 'id=\"random.weibullvariate', 'id=\"random.Random', 'id=\"random.SystemRandom']\n"
     ]
    }
   ],
   "source": [
    "# Find all the information we're looking for with regex\n",
    "# In this case, it would be every string at starts with id='random.'\n",
    "\n",
    "function_names = re.findall('id=\"random.\\w+' , str(names)) # '\\w+' which means the string should end with the function name\n",
    "\n",
    "# Let print the results\n",
    "print(function_names)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "**We are almost there! We just need to remove the first few characters from each string.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['random.seed', 'random.getstate', 'random.setstate', 'random.randbytes', 'random.randrange', 'random.randint', 'random.getrandbits', 'random.choice', 'random.choices', 'random.shuffle', 'random.sample', 'random.random', 'random.uniform', 'random.triangular', 'random.betavariate', 'random.expovariate', 'random.gammavariate', 'random.gauss', 'random.lognormvariate', 'random.normalvariate', 'random.vonmisesvariate', 'random.paretovariate', 'random.weibullvariate', 'random.Random', 'random.SystemRandom']\n"
     ]
    }
   ],
   "source": [
    "# Using list comprehension to edit our values:\n",
    "\n",
    "function_names = [item[4:] for item in function_names]\n",
    "\n",
    "# Let print the results\n",
    "print(function_names)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Perfect! Now we need to do the same with the function description.\n",
    "We have to target the description details with tag - `dd`**\n",
    "\n",
    "<br>\n",
    "\n",
    "\n",
    "<img src = \"https://docs.google.com/uc?export=download&id=169-W93jfnmbwHejyP4QV4sDQm9LgriwB\" />\n",
    "\n",
    "<br>\n",
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Find all the function description\n",
    "\n",
    "description = soup.body.findAll('dd')\n",
    "\n",
    "#print(description)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Wow it looks very complicated! There are lots of tags here (`<em>` tags). These unnecessary elements from the above method would take a long time to get rid of manually.**\n",
    "    \n",
    "Luckily, BeautifulSoup is not only beautiful, it's also smart. Let's look at the `.text` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a list\n",
    "\n",
    "function_usage = []\n",
    "\n",
    "# Create a loop\n",
    "\n",
    "for item in description:\n",
    "    item = item.text      #  Save the extracted text to a variable\n",
    "    item = item.replace('\\n', ' ')     # to get rid of the next line operator which is `\\n` \n",
    "    function_usage.append(item)\n",
    "    \n",
    "#print(function_usage)  # Don't get overwhelmed! they are just all the function description from the above function names"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Length of function_names: 25\n",
      " Length of function_usage: 25\n"
     ]
    }
   ],
   "source": [
    "# Let's check the length of the function_names and function_usage\n",
    "\n",
    "print(f' Length of function_names: {len(function_names)}')\n",
    "print(f' Length of function_usage: {len(function_usage)}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Make A Database"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>function name</th>\n",
       "      <th>function usage</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>random.seed</td>\n",
       "      <td>Initialize the random number generator.If a is...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>random.getstate</td>\n",
       "      <td>Return an object capturing the current interna...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>random.setstate</td>\n",
       "      <td>state should have been obtained from a previou...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>random.randbytes</td>\n",
       "      <td>Generate n random bytes.This method should not...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>random.randrange</td>\n",
       "      <td>Return a randomly selected element from range(...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>random.randint</td>\n",
       "      <td>Return a random integer N such that a &lt;= N &lt;= ...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>random.getrandbits</td>\n",
       "      <td>Returns a non-negative Python integer with k r...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>random.choice</td>\n",
       "      <td>Return a random element from the non-empty seq...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>random.choices</td>\n",
       "      <td>Return a k sized list of elements chosen from ...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>random.shuffle</td>\n",
       "      <td>Shuffle the sequence x in place.The optional a...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>random.sample</td>\n",
       "      <td>Return a k length list of unique elements chos...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>random.random</td>\n",
       "      <td>Return the next random floating point number i...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>random.uniform</td>\n",
       "      <td>Return a random floating point number N such t...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>random.triangular</td>\n",
       "      <td>Return a random floating point number N such t...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>random.betavariate</td>\n",
       "      <td>Beta distribution.  Conditions on the paramete...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>random.expovariate</td>\n",
       "      <td>Exponential distribution.  lambd is 1.0 divide...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>random.gammavariate</td>\n",
       "      <td>Gamma distribution.  (Not the gamma function!)...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>random.gauss</td>\n",
       "      <td>Gaussian distribution.  mu is the mean, and si...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>random.lognormvariate</td>\n",
       "      <td>Log normal distribution.  If you take the natu...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>random.normalvariate</td>\n",
       "      <td>Normal distribution.  mu is the mean, and sigm...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>random.vonmisesvariate</td>\n",
       "      <td>mu is the mean angle, expressed in radians bet...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>random.paretovariate</td>\n",
       "      <td>Pareto distribution.  alpha is the shape param...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>random.weibullvariate</td>\n",
       "      <td>Weibull distribution.  alpha is the scale para...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>random.Random</td>\n",
       "      <td>Class that implements the default pseudo-rando...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>random.SystemRandom</td>\n",
       "      <td>Class that uses the os.urandom() function for ...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             function name                                     function usage\n",
       "0              random.seed  Initialize the random number generator.If a is...\n",
       "1          random.getstate  Return an object capturing the current interna...\n",
       "2          random.setstate  state should have been obtained from a previou...\n",
       "3         random.randbytes  Generate n random bytes.This method should not...\n",
       "4         random.randrange  Return a randomly selected element from range(...\n",
       "5           random.randint  Return a random integer N such that a <= N <= ...\n",
       "6       random.getrandbits  Returns a non-negative Python integer with k r...\n",
       "7            random.choice  Return a random element from the non-empty seq...\n",
       "8           random.choices  Return a k sized list of elements chosen from ...\n",
       "9           random.shuffle  Shuffle the sequence x in place.The optional a...\n",
       "10           random.sample  Return a k length list of unique elements chos...\n",
       "11           random.random  Return the next random floating point number i...\n",
       "12          random.uniform  Return a random floating point number N such t...\n",
       "13       random.triangular  Return a random floating point number N such t...\n",
       "14      random.betavariate  Beta distribution.  Conditions on the paramete...\n",
       "15      random.expovariate  Exponential distribution.  lambd is 1.0 divide...\n",
       "16     random.gammavariate  Gamma distribution.  (Not the gamma function!)...\n",
       "17            random.gauss  Gaussian distribution.  mu is the mean, and si...\n",
       "18   random.lognormvariate  Log normal distribution.  If you take the natu...\n",
       "19    random.normalvariate  Normal distribution.  mu is the mean, and sigm...\n",
       "20  random.vonmisesvariate  mu is the mean angle, expressed in radians bet...\n",
       "21    random.paretovariate  Pareto distribution.  alpha is the shape param...\n",
       "22   random.weibullvariate  Weibull distribution.  alpha is the scale para...\n",
       "23           random.Random  Class that implements the default pseudo-rando...\n",
       "24     random.SystemRandom  Class that uses the os.urandom() function for ..."
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a dataframe since the length of both variables are equal!\n",
    "\n",
    "data = pd.DataFrame( {  'function name': function_names, \n",
    "                      'function usage' : function_usage  } )\n",
    "\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# Let make a CSV file from the dataframe\n",
    "\n",
    "data.to_csv('random_function.csv')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**BONUS: if you want to target a specific attributes, for example `id=\"bookeeping-functions\"` you can use the following code:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Target specific attributes\n",
    "\n",
    "#example = soup.body.findAll ('div', attrs = {'id' : 'bookeeping-functions'})\n",
    "#print(example)    # you can get very specific result with BeautifulSoup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## II. Web Scraping Using Pandas "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pandas is very useful! We can easily scrape data using the pandas `read_html()` function for your data science project. \n",
    "\n",
    "We will be web scraping NBA player stats data and perform a quick data exploration from the website [basketball-reference.com](https://www.basketball-reference.com)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Get The URL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we want to check out the specific [URL](https://www.basketball-reference.com/leagues/NBA_2020_per_game.html) that we are going to scrape the data - the NBA Player Stats of season 2019-2020.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'https://www.basketball-reference.com/leagues/NBA_2020_per_game.html'"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Method 1: only 1 year\n",
    "\n",
    "# URL of the player stats in 2020\n",
    "\n",
    "url = 'https://www.basketball-reference.com/leagues/NBA_2020_per_game.html'\n",
    "url"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "https://www.basketball-reference.com/leagues/NBA_2016_per_game.html\n",
      "https://www.basketball-reference.com/leagues/NBA_2017_per_game.html\n",
      "https://www.basketball-reference.com/leagues/NBA_2018_per_game.html\n",
      "https://www.basketball-reference.com/leagues/NBA_2019_per_game.html\n",
      "https://www.basketball-reference.com/leagues/NBA_2020_per_game.html\n"
     ]
    }
   ],
   "source": [
    "# Method 2: multiple years\n",
    "\n",
    "years = ['2016', '2017', '2018', '2019', '2020']\n",
    "str = 'https://www.basketball-reference.com/leagues/NBA_{}_per_game.html'\n",
    "\n",
    "for year in years:\n",
    "    url = str.format(year)\n",
    "    print(url)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Read The HTML Webpage Into Pandas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[      Rk                    Player Pos Age   Tm   G  GS    MP   FG   FGA  ...  \\\n",
      "0      1              Steven Adams   C  26  OKC  63  63  26.7  4.5   7.6  ...   \n",
      "1      2               Bam Adebayo  PF  22  MIA  72  72  33.6  6.1  11.0  ...   \n",
      "2      3         LaMarcus Aldridge   C  34  SAS  53  53  33.1  7.4  15.0  ...   \n",
      "3      4            Kyle Alexander   C  23  MIA   2   0   6.5  0.5   1.0  ...   \n",
      "4      5  Nickeil Alexander-Walker  SG  21  NOP  47   1  12.6  2.1   5.7  ...   \n",
      "..   ...                       ...  ..  ..  ...  ..  ..   ...  ...   ...  ...   \n",
      "672  525                Trae Young  PG  21  ATL  60  60  35.3  9.1  20.8  ...   \n",
      "673  526               Cody Zeller   C  27  CHO  58  39  23.1  4.3   8.3  ...   \n",
      "674  527              Tyler Zeller   C  30  SAS   2   0   2.0  0.5   2.0  ...   \n",
      "675  528                Ante Žižić   C  23  CLE  22   0  10.0  1.9   3.3  ...   \n",
      "676  529               Ivica Zubac   C  22  LAC  72  70  18.4  3.3   5.3  ...   \n",
      "\n",
      "      FT%  ORB  DRB   TRB  AST  STL  BLK  TOV   PF   PTS  \n",
      "0    .582  3.3  6.0   9.3  2.3  0.8  1.1  1.5  1.9  10.9  \n",
      "1    .691  2.4  7.8  10.2  5.1  1.1  1.3  2.8  2.5  15.9  \n",
      "2    .827  1.9  5.5   7.4  2.4  0.7  1.6  1.4  2.4  18.9  \n",
      "3     NaN  1.0  0.5   1.5  0.0  0.0  0.0  0.5  0.5   1.0  \n",
      "4    .676  0.2  1.6   1.8  1.9  0.4  0.2  1.1  1.2   5.7  \n",
      "..    ...  ...  ...   ...  ...  ...  ...  ...  ...   ...  \n",
      "672  .860  0.5  3.7   4.3  9.3  1.1  0.1  4.8  1.7  29.6  \n",
      "673  .682  2.8  4.3   7.1  1.5  0.7  0.4  1.3  2.4  11.1  \n",
      "674   NaN  1.5  0.5   2.0  0.0  0.0  0.0  0.0  0.0   1.0  \n",
      "675  .737  0.8  2.2   3.0  0.3  0.3  0.2  0.5  1.2   4.4  \n",
      "676  .747  2.7  4.8   7.5  1.1  0.2  0.9  0.8  2.3   8.3  \n",
      "\n",
      "[677 rows x 30 columns]]\n"
     ]
    }
   ],
   "source": [
    "# Let check URL of the player stats in 2020\n",
    "\n",
    "url = 'https://www.basketball-reference.com/leagues/NBA_2020_per_game.html'\n",
    "\n",
    "# Using pd.read_html()\n",
    "\n",
    "df = pd.read_html(url, header = 0)\n",
    "\n",
    "print(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**It looks a little bit messy. What we actually have here is a list of DataFrames. We can beautify this object using Pandas (without any additional libraries!)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "number of tables in df: 1\n",
      "================\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Rk</th>\n",
       "      <th>Player</th>\n",
       "      <th>Pos</th>\n",
       "      <th>Age</th>\n",
       "      <th>Tm</th>\n",
       "      <th>G</th>\n",
       "      <th>GS</th>\n",
       "      <th>MP</th>\n",
       "      <th>FG</th>\n",
       "      <th>FGA</th>\n",
       "      <th>...</th>\n",
       "      <th>FT%</th>\n",
       "      <th>ORB</th>\n",
       "      <th>DRB</th>\n",
       "      <th>TRB</th>\n",
       "      <th>AST</th>\n",
       "      <th>STL</th>\n",
       "      <th>BLK</th>\n",
       "      <th>TOV</th>\n",
       "      <th>PF</th>\n",
       "      <th>PTS</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>Steven Adams</td>\n",
       "      <td>C</td>\n",
       "      <td>26</td>\n",
       "      <td>OKC</td>\n",
       "      <td>63</td>\n",
       "      <td>63</td>\n",
       "      <td>26.7</td>\n",
       "      <td>4.5</td>\n",
       "      <td>7.6</td>\n",
       "      <td>...</td>\n",
       "      <td>.582</td>\n",
       "      <td>3.3</td>\n",
       "      <td>6.0</td>\n",
       "      <td>9.3</td>\n",
       "      <td>2.3</td>\n",
       "      <td>0.8</td>\n",
       "      <td>1.1</td>\n",
       "      <td>1.5</td>\n",
       "      <td>1.9</td>\n",
       "      <td>10.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>Bam Adebayo</td>\n",
       "      <td>PF</td>\n",
       "      <td>22</td>\n",
       "      <td>MIA</td>\n",
       "      <td>72</td>\n",
       "      <td>72</td>\n",
       "      <td>33.6</td>\n",
       "      <td>6.1</td>\n",
       "      <td>11.0</td>\n",
       "      <td>...</td>\n",
       "      <td>.691</td>\n",
       "      <td>2.4</td>\n",
       "      <td>7.8</td>\n",
       "      <td>10.2</td>\n",
       "      <td>5.1</td>\n",
       "      <td>1.1</td>\n",
       "      <td>1.3</td>\n",
       "      <td>2.8</td>\n",
       "      <td>2.5</td>\n",
       "      <td>15.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>LaMarcus Aldridge</td>\n",
       "      <td>C</td>\n",
       "      <td>34</td>\n",
       "      <td>SAS</td>\n",
       "      <td>53</td>\n",
       "      <td>53</td>\n",
       "      <td>33.1</td>\n",
       "      <td>7.4</td>\n",
       "      <td>15.0</td>\n",
       "      <td>...</td>\n",
       "      <td>.827</td>\n",
       "      <td>1.9</td>\n",
       "      <td>5.5</td>\n",
       "      <td>7.4</td>\n",
       "      <td>2.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>2.4</td>\n",
       "      <td>18.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>Kyle Alexander</td>\n",
       "      <td>C</td>\n",
       "      <td>23</td>\n",
       "      <td>MIA</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>6.5</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.0</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>Nickeil Alexander-Walker</td>\n",
       "      <td>SG</td>\n",
       "      <td>21</td>\n",
       "      <td>NOP</td>\n",
       "      <td>47</td>\n",
       "      <td>1</td>\n",
       "      <td>12.6</td>\n",
       "      <td>2.1</td>\n",
       "      <td>5.7</td>\n",
       "      <td>...</td>\n",
       "      <td>.676</td>\n",
       "      <td>0.2</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.8</td>\n",
       "      <td>1.9</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>1.1</td>\n",
       "      <td>1.2</td>\n",
       "      <td>5.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "      <td>Grayson Allen</td>\n",
       "      <td>SG</td>\n",
       "      <td>24</td>\n",
       "      <td>MEM</td>\n",
       "      <td>38</td>\n",
       "      <td>0</td>\n",
       "      <td>18.9</td>\n",
       "      <td>3.1</td>\n",
       "      <td>6.6</td>\n",
       "      <td>...</td>\n",
       "      <td>.867</td>\n",
       "      <td>0.2</td>\n",
       "      <td>2.0</td>\n",
       "      <td>2.2</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.3</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.9</td>\n",
       "      <td>1.4</td>\n",
       "      <td>8.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>7</td>\n",
       "      <td>Jarrett Allen</td>\n",
       "      <td>C</td>\n",
       "      <td>21</td>\n",
       "      <td>BRK</td>\n",
       "      <td>70</td>\n",
       "      <td>64</td>\n",
       "      <td>26.5</td>\n",
       "      <td>4.3</td>\n",
       "      <td>6.6</td>\n",
       "      <td>...</td>\n",
       "      <td>.633</td>\n",
       "      <td>3.1</td>\n",
       "      <td>6.5</td>\n",
       "      <td>9.6</td>\n",
       "      <td>1.6</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.3</td>\n",
       "      <td>1.1</td>\n",
       "      <td>2.3</td>\n",
       "      <td>11.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8</td>\n",
       "      <td>Kadeem Allen</td>\n",
       "      <td>PG</td>\n",
       "      <td>27</td>\n",
       "      <td>NYK</td>\n",
       "      <td>10</td>\n",
       "      <td>0</td>\n",
       "      <td>11.7</td>\n",
       "      <td>1.9</td>\n",
       "      <td>4.4</td>\n",
       "      <td>...</td>\n",
       "      <td>.636</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.7</td>\n",
       "      <td>0.9</td>\n",
       "      <td>2.1</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.7</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>9</td>\n",
       "      <td>Al-Farouq Aminu</td>\n",
       "      <td>PF</td>\n",
       "      <td>29</td>\n",
       "      <td>ORL</td>\n",
       "      <td>18</td>\n",
       "      <td>2</td>\n",
       "      <td>21.1</td>\n",
       "      <td>1.4</td>\n",
       "      <td>4.8</td>\n",
       "      <td>...</td>\n",
       "      <td>.655</td>\n",
       "      <td>1.3</td>\n",
       "      <td>3.5</td>\n",
       "      <td>4.8</td>\n",
       "      <td>1.2</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.9</td>\n",
       "      <td>1.5</td>\n",
       "      <td>4.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>10</td>\n",
       "      <td>Justin Anderson</td>\n",
       "      <td>SG</td>\n",
       "      <td>26</td>\n",
       "      <td>BRK</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "      <td>10.7</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.8</td>\n",
       "      <td>...</td>\n",
       "      <td>.500</td>\n",
       "      <td>0.1</td>\n",
       "      <td>2.0</td>\n",
       "      <td>2.1</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.4</td>\n",
       "      <td>1.3</td>\n",
       "      <td>2.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>11</td>\n",
       "      <td>Kyle Anderson</td>\n",
       "      <td>SF</td>\n",
       "      <td>26</td>\n",
       "      <td>MEM</td>\n",
       "      <td>67</td>\n",
       "      <td>28</td>\n",
       "      <td>19.9</td>\n",
       "      <td>2.3</td>\n",
       "      <td>4.9</td>\n",
       "      <td>...</td>\n",
       "      <td>.667</td>\n",
       "      <td>0.9</td>\n",
       "      <td>3.4</td>\n",
       "      <td>4.3</td>\n",
       "      <td>2.4</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.7</td>\n",
       "      <td>5.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>12</td>\n",
       "      <td>Ryan Anderson</td>\n",
       "      <td>C</td>\n",
       "      <td>31</td>\n",
       "      <td>HOU</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.5</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3.5</td>\n",
       "      <td>3.5</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.5</td>\n",
       "      <td>2.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>13</td>\n",
       "      <td>Giannis Antetokounmpo</td>\n",
       "      <td>PF</td>\n",
       "      <td>25</td>\n",
       "      <td>MIL</td>\n",
       "      <td>63</td>\n",
       "      <td>63</td>\n",
       "      <td>30.4</td>\n",
       "      <td>10.9</td>\n",
       "      <td>19.7</td>\n",
       "      <td>...</td>\n",
       "      <td>.633</td>\n",
       "      <td>2.2</td>\n",
       "      <td>11.4</td>\n",
       "      <td>13.6</td>\n",
       "      <td>5.6</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.7</td>\n",
       "      <td>3.1</td>\n",
       "      <td>29.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>14</td>\n",
       "      <td>Kostas Antetokounmpo</td>\n",
       "      <td>PF</td>\n",
       "      <td>22</td>\n",
       "      <td>LAL</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.6</td>\n",
       "      <td>...</td>\n",
       "      <td>.500</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.4</td>\n",
       "      <td>1.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>15</td>\n",
       "      <td>Thanasis Antetokounmpo</td>\n",
       "      <td>SF</td>\n",
       "      <td>27</td>\n",
       "      <td>MIL</td>\n",
       "      <td>20</td>\n",
       "      <td>2</td>\n",
       "      <td>6.5</td>\n",
       "      <td>1.2</td>\n",
       "      <td>2.4</td>\n",
       "      <td>...</td>\n",
       "      <td>.412</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.2</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.9</td>\n",
       "      <td>2.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>16</td>\n",
       "      <td>Carmelo Anthony</td>\n",
       "      <td>PF</td>\n",
       "      <td>35</td>\n",
       "      <td>POR</td>\n",
       "      <td>58</td>\n",
       "      <td>58</td>\n",
       "      <td>32.8</td>\n",
       "      <td>5.8</td>\n",
       "      <td>13.5</td>\n",
       "      <td>...</td>\n",
       "      <td>.845</td>\n",
       "      <td>1.2</td>\n",
       "      <td>5.1</td>\n",
       "      <td>6.3</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.7</td>\n",
       "      <td>2.9</td>\n",
       "      <td>15.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>17</td>\n",
       "      <td>OG Anunoby</td>\n",
       "      <td>SF</td>\n",
       "      <td>22</td>\n",
       "      <td>TOR</td>\n",
       "      <td>69</td>\n",
       "      <td>68</td>\n",
       "      <td>29.9</td>\n",
       "      <td>4.1</td>\n",
       "      <td>8.2</td>\n",
       "      <td>...</td>\n",
       "      <td>.706</td>\n",
       "      <td>1.2</td>\n",
       "      <td>4.1</td>\n",
       "      <td>5.3</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>1.1</td>\n",
       "      <td>2.4</td>\n",
       "      <td>10.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>18</td>\n",
       "      <td>Ryan Arcidiacono</td>\n",
       "      <td>PG</td>\n",
       "      <td>25</td>\n",
       "      <td>CHI</td>\n",
       "      <td>58</td>\n",
       "      <td>4</td>\n",
       "      <td>16.0</td>\n",
       "      <td>1.6</td>\n",
       "      <td>3.8</td>\n",
       "      <td>...</td>\n",
       "      <td>.711</td>\n",
       "      <td>0.3</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.9</td>\n",
       "      <td>1.7</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.7</td>\n",
       "      <td>4.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>19</td>\n",
       "      <td>Trevor Ariza</td>\n",
       "      <td>SF</td>\n",
       "      <td>34</td>\n",
       "      <td>TOT</td>\n",
       "      <td>53</td>\n",
       "      <td>21</td>\n",
       "      <td>28.2</td>\n",
       "      <td>2.7</td>\n",
       "      <td>6.1</td>\n",
       "      <td>...</td>\n",
       "      <td>.838</td>\n",
       "      <td>0.6</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.6</td>\n",
       "      <td>1.7</td>\n",
       "      <td>1.3</td>\n",
       "      <td>0.3</td>\n",
       "      <td>1.1</td>\n",
       "      <td>2.1</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>19</td>\n",
       "      <td>Trevor Ariza</td>\n",
       "      <td>SF</td>\n",
       "      <td>34</td>\n",
       "      <td>SAC</td>\n",
       "      <td>32</td>\n",
       "      <td>0</td>\n",
       "      <td>24.7</td>\n",
       "      <td>2.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>...</td>\n",
       "      <td>.778</td>\n",
       "      <td>0.7</td>\n",
       "      <td>3.9</td>\n",
       "      <td>4.6</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.1</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.9</td>\n",
       "      <td>2.0</td>\n",
       "      <td>6.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>19</td>\n",
       "      <td>Trevor Ariza</td>\n",
       "      <td>SF</td>\n",
       "      <td>34</td>\n",
       "      <td>POR</td>\n",
       "      <td>21</td>\n",
       "      <td>21</td>\n",
       "      <td>33.4</td>\n",
       "      <td>3.7</td>\n",
       "      <td>7.6</td>\n",
       "      <td>...</td>\n",
       "      <td>.872</td>\n",
       "      <td>0.6</td>\n",
       "      <td>4.1</td>\n",
       "      <td>4.8</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1.6</td>\n",
       "      <td>0.4</td>\n",
       "      <td>1.3</td>\n",
       "      <td>2.3</td>\n",
       "      <td>11.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>20</td>\n",
       "      <td>D.J. Augustin</td>\n",
       "      <td>PG</td>\n",
       "      <td>32</td>\n",
       "      <td>ORL</td>\n",
       "      <td>57</td>\n",
       "      <td>13</td>\n",
       "      <td>24.9</td>\n",
       "      <td>3.2</td>\n",
       "      <td>8.1</td>\n",
       "      <td>...</td>\n",
       "      <td>.890</td>\n",
       "      <td>0.4</td>\n",
       "      <td>1.8</td>\n",
       "      <td>2.1</td>\n",
       "      <td>4.6</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.5</td>\n",
       "      <td>1.3</td>\n",
       "      <td>10.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>Rk</td>\n",
       "      <td>Player</td>\n",
       "      <td>Pos</td>\n",
       "      <td>Age</td>\n",
       "      <td>Tm</td>\n",
       "      <td>G</td>\n",
       "      <td>GS</td>\n",
       "      <td>MP</td>\n",
       "      <td>FG</td>\n",
       "      <td>FGA</td>\n",
       "      <td>...</td>\n",
       "      <td>FT%</td>\n",
       "      <td>ORB</td>\n",
       "      <td>DRB</td>\n",
       "      <td>TRB</td>\n",
       "      <td>AST</td>\n",
       "      <td>STL</td>\n",
       "      <td>BLK</td>\n",
       "      <td>TOV</td>\n",
       "      <td>PF</td>\n",
       "      <td>PTS</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>21</td>\n",
       "      <td>Deandre Ayton</td>\n",
       "      <td>C</td>\n",
       "      <td>21</td>\n",
       "      <td>PHO</td>\n",
       "      <td>38</td>\n",
       "      <td>32</td>\n",
       "      <td>32.5</td>\n",
       "      <td>8.2</td>\n",
       "      <td>14.9</td>\n",
       "      <td>...</td>\n",
       "      <td>.753</td>\n",
       "      <td>3.9</td>\n",
       "      <td>7.6</td>\n",
       "      <td>11.5</td>\n",
       "      <td>1.9</td>\n",
       "      <td>0.7</td>\n",
       "      <td>1.5</td>\n",
       "      <td>2.1</td>\n",
       "      <td>3.1</td>\n",
       "      <td>18.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>22</td>\n",
       "      <td>Dwayne Bacon</td>\n",
       "      <td>SG</td>\n",
       "      <td>24</td>\n",
       "      <td>CHO</td>\n",
       "      <td>39</td>\n",
       "      <td>11</td>\n",
       "      <td>17.6</td>\n",
       "      <td>2.2</td>\n",
       "      <td>6.3</td>\n",
       "      <td>...</td>\n",
       "      <td>.660</td>\n",
       "      <td>0.4</td>\n",
       "      <td>2.2</td>\n",
       "      <td>2.6</td>\n",
       "      <td>1.3</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.9</td>\n",
       "      <td>1.3</td>\n",
       "      <td>5.7</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>25 rows × 30 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    Rk                    Player  Pos  Age   Tm   G  GS    MP    FG   FGA  \\\n",
       "0    1              Steven Adams    C   26  OKC  63  63  26.7   4.5   7.6   \n",
       "1    2               Bam Adebayo   PF   22  MIA  72  72  33.6   6.1  11.0   \n",
       "2    3         LaMarcus Aldridge    C   34  SAS  53  53  33.1   7.4  15.0   \n",
       "3    4            Kyle Alexander    C   23  MIA   2   0   6.5   0.5   1.0   \n",
       "4    5  Nickeil Alexander-Walker   SG   21  NOP  47   1  12.6   2.1   5.7   \n",
       "5    6             Grayson Allen   SG   24  MEM  38   0  18.9   3.1   6.6   \n",
       "6    7             Jarrett Allen    C   21  BRK  70  64  26.5   4.3   6.6   \n",
       "7    8              Kadeem Allen   PG   27  NYK  10   0  11.7   1.9   4.4   \n",
       "8    9           Al-Farouq Aminu   PF   29  ORL  18   2  21.1   1.4   4.8   \n",
       "9   10           Justin Anderson   SG   26  BRK  10   1  10.7   1.0   3.8   \n",
       "10  11             Kyle Anderson   SF   26  MEM  67  28  19.9   2.3   4.9   \n",
       "11  12             Ryan Anderson    C   31  HOU   2   0   7.0   1.0   3.5   \n",
       "12  13     Giannis Antetokounmpo   PF   25  MIL  63  63  30.4  10.9  19.7   \n",
       "13  14      Kostas Antetokounmpo   PF   22  LAL   5   0   4.0   0.6   0.6   \n",
       "14  15    Thanasis Antetokounmpo   SF   27  MIL  20   2   6.5   1.2   2.4   \n",
       "15  16           Carmelo Anthony   PF   35  POR  58  58  32.8   5.8  13.5   \n",
       "16  17                OG Anunoby   SF   22  TOR  69  68  29.9   4.1   8.2   \n",
       "17  18          Ryan Arcidiacono   PG   25  CHI  58   4  16.0   1.6   3.8   \n",
       "18  19              Trevor Ariza   SF   34  TOT  53  21  28.2   2.7   6.1   \n",
       "19  19              Trevor Ariza   SF   34  SAC  32   0  24.7   2.0   5.2   \n",
       "20  19              Trevor Ariza   SF   34  POR  21  21  33.4   3.7   7.6   \n",
       "21  20             D.J. Augustin   PG   32  ORL  57  13  24.9   3.2   8.1   \n",
       "22  Rk                    Player  Pos  Age   Tm   G  GS    MP    FG   FGA   \n",
       "23  21             Deandre Ayton    C   21  PHO  38  32  32.5   8.2  14.9   \n",
       "24  22              Dwayne Bacon   SG   24  CHO  39  11  17.6   2.2   6.3   \n",
       "\n",
       "    ...   FT%  ORB   DRB   TRB  AST  STL  BLK  TOV   PF   PTS  \n",
       "0   ...  .582  3.3   6.0   9.3  2.3  0.8  1.1  1.5  1.9  10.9  \n",
       "1   ...  .691  2.4   7.8  10.2  5.1  1.1  1.3  2.8  2.5  15.9  \n",
       "2   ...  .827  1.9   5.5   7.4  2.4  0.7  1.6  1.4  2.4  18.9  \n",
       "3   ...   NaN  1.0   0.5   1.5  0.0  0.0  0.0  0.5  0.5   1.0  \n",
       "4   ...  .676  0.2   1.6   1.8  1.9  0.4  0.2  1.1  1.2   5.7  \n",
       "5   ...  .867  0.2   2.0   2.2  1.4  0.3  0.1  0.9  1.4   8.7  \n",
       "6   ...  .633  3.1   6.5   9.6  1.6  0.6  1.3  1.1  2.3  11.1  \n",
       "7   ...  .636  0.2   0.7   0.9  2.1  0.5  0.2  0.8  0.7   5.0  \n",
       "8   ...  .655  1.3   3.5   4.8  1.2  1.0  0.4  0.9  1.5   4.3  \n",
       "9   ...  .500  0.1   2.0   2.1  0.8  0.0  0.6  0.4  1.3   2.8  \n",
       "10  ...  .667  0.9   3.4   4.3  2.4  0.8  0.6  1.0  1.7   5.8  \n",
       "11  ...   NaN  0.0   3.5   3.5  1.0  0.5  0.0  0.5  0.5   2.5  \n",
       "12  ...  .633  2.2  11.4  13.6  5.6  1.0  1.0  3.7  3.1  29.5  \n",
       "13  ...  .500  0.4   0.2   0.6  0.4  0.0  0.0  0.2  0.4   1.4  \n",
       "14  ...  .412  0.6   0.6   1.2  0.8  0.4  0.1  0.6  0.9   2.8  \n",
       "15  ...  .845  1.2   5.1   6.3  1.5  0.8  0.5  1.7  2.9  15.4  \n",
       "16  ...  .706  1.2   4.1   5.3  1.6  1.4  0.7  1.1  2.4  10.6  \n",
       "17  ...  .711  0.3   1.6   1.9  1.7  0.5  0.1  0.6  1.7   4.5  \n",
       "18  ...  .838  0.6   4.0   4.6  1.7  1.3  0.3  1.1  2.1   8.0  \n",
       "19  ...  .778  0.7   3.9   4.6  1.6  1.1  0.2  0.9  2.0   6.0  \n",
       "20  ...  .872  0.6   4.1   4.8  2.0  1.6  0.4  1.3  2.3  11.0  \n",
       "21  ...  .890  0.4   1.8   2.1  4.6  0.6  0.0  1.5  1.3  10.5  \n",
       "22  ...   FT%  ORB   DRB   TRB  AST  STL  BLK  TOV   PF   PTS  \n",
       "23  ...  .753  3.9   7.6  11.5  1.9  0.7  1.5  2.1  3.1  18.2  \n",
       "24  ...  .660  0.4   2.2   2.6  1.3  0.6  0.1  0.9  1.3   5.7  \n",
       "\n",
       "[25 rows x 30 columns]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check number of DataFrames in this list\n",
    "\n",
    "print(f'number of tables in df: {len(df)}') \n",
    "\n",
    "print('================')\n",
    "\n",
    "# Since there is only 1, pull out the 0th element:\n",
    "df[0].head(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wow! You'll notice that there are some missing values (NaN) and multiple occurences of some player names because they have been a part of different teams in the same year."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data Cleaning"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**We can see on the website that the header repeats itself in every 20 players. We'll have to remove the subsequent headers and keep only the first header:**\n",
    "\n",
    "<br>\n",
    "<br>\n",
    "<img src = \"https://docs.google.com/uc?export=download&id=1CEQs7TNFr4Nak0sQK10QYXl06uXcvrLN\" />\n",
    "\n",
    "<br>\n",
    "<br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Rk</th>\n",
       "      <th>Player</th>\n",
       "      <th>Pos</th>\n",
       "      <th>Age</th>\n",
       "      <th>Tm</th>\n",
       "      <th>G</th>\n",
       "      <th>GS</th>\n",
       "      <th>MP</th>\n",
       "      <th>FG</th>\n",
       "      <th>FGA</th>\n",
       "      <th>...</th>\n",
       "      <th>FT%</th>\n",
       "      <th>ORB</th>\n",
       "      <th>DRB</th>\n",
       "      <th>TRB</th>\n",
       "      <th>AST</th>\n",
       "      <th>STL</th>\n",
       "      <th>BLK</th>\n",
       "      <th>TOV</th>\n",
       "      <th>PF</th>\n",
       "      <th>PTS</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>Rk</td>\n",
       "      <td>Player</td>\n",
       "      <td>Pos</td>\n",
       "      <td>Age</td>\n",
       "      <td>Tm</td>\n",
       "      <td>G</td>\n",
       "      <td>GS</td>\n",
       "      <td>MP</td>\n",
       "      <td>FG</td>\n",
       "      <td>FGA</td>\n",
       "      <td>...</td>\n",
       "      <td>FT%</td>\n",
       "      <td>ORB</td>\n",
       "      <td>DRB</td>\n",
       "      <td>TRB</td>\n",
       "      <td>AST</td>\n",
       "      <td>STL</td>\n",
       "      <td>BLK</td>\n",
       "      <td>TOV</td>\n",
       "      <td>PF</td>\n",
       "      <td>PTS</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>53</th>\n",
       "      <td>Rk</td>\n",
       "      <td>Player</td>\n",
       "      <td>Pos</td>\n",
       "      <td>Age</td>\n",
       "      <td>Tm</td>\n",
       "      <td>G</td>\n",
       "      <td>GS</td>\n",
       "      <td>MP</td>\n",
       "      <td>FG</td>\n",
       "      <td>FGA</td>\n",
       "      <td>...</td>\n",
       "      <td>FT%</td>\n",
       "      <td>ORB</td>\n",
       "      <td>DRB</td>\n",
       "      <td>TRB</td>\n",
       "      <td>AST</td>\n",
       "      <td>STL</td>\n",
       "      <td>BLK</td>\n",
       "      <td>TOV</td>\n",
       "      <td>PF</td>\n",
       "      <td>PTS</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>Rk</td>\n",
       "      <td>Player</td>\n",
       "      <td>Pos</td>\n",
       "      <td>Age</td>\n",
       "      <td>Tm</td>\n",
       "      <td>G</td>\n",
       "      <td>GS</td>\n",
       "      <td>MP</td>\n",
       "      <td>FG</td>\n",
       "      <td>FGA</td>\n",
       "      <td>...</td>\n",
       "      <td>FT%</td>\n",
       "      <td>ORB</td>\n",
       "      <td>DRB</td>\n",
       "      <td>TRB</td>\n",
       "      <td>AST</td>\n",
       "      <td>STL</td>\n",
       "      <td>BLK</td>\n",
       "      <td>TOV</td>\n",
       "      <td>PF</td>\n",
       "      <td>PTS</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>101</th>\n",
       "      <td>Rk</td>\n",
       "      <td>Player</td>\n",
       "      <td>Pos</td>\n",
       "      <td>Age</td>\n",
       "      <td>Tm</td>\n",
       "      <td>G</td>\n",
       "      <td>GS</td>\n",
       "      <td>MP</td>\n",
       "      <td>FG</td>\n",
       "      <td>FGA</td>\n",
       "      <td>...</td>\n",
       "      <td>FT%</td>\n",
       "      <td>ORB</td>\n",
       "      <td>DRB</td>\n",
       "      <td>TRB</td>\n",
       "      <td>AST</td>\n",
       "      <td>STL</td>\n",
       "      <td>BLK</td>\n",
       "      <td>TOV</td>\n",
       "      <td>PF</td>\n",
       "      <td>PTS</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>130</th>\n",
       "      <td>Rk</td>\n",
       "      <td>Player</td>\n",
       "      <td>Pos</td>\n",
       "      <td>Age</td>\n",
       "      <td>Tm</td>\n",
       "      <td>G</td>\n",
       "      <td>GS</td>\n",
       "      <td>MP</td>\n",
       "      <td>FG</td>\n",
       "      <td>FGA</td>\n",
       "      <td>...</td>\n",
       "      <td>FT%</td>\n",
       "      <td>ORB</td>\n",
       "      <td>DRB</td>\n",
       "      <td>TRB</td>\n",
       "      <td>AST</td>\n",
       "      <td>STL</td>\n",
       "      <td>BLK</td>\n",
       "      <td>TOV</td>\n",
       "      <td>PF</td>\n",
       "      <td>PTS</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 30 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "     Rk  Player  Pos  Age  Tm  G  GS  MP  FG  FGA  ...  FT%  ORB  DRB  TRB  \\\n",
       "22   Rk  Player  Pos  Age  Tm  G  GS  MP  FG  FGA  ...  FT%  ORB  DRB  TRB   \n",
       "53   Rk  Player  Pos  Age  Tm  G  GS  MP  FG  FGA  ...  FT%  ORB  DRB  TRB   \n",
       "76   Rk  Player  Pos  Age  Tm  G  GS  MP  FG  FGA  ...  FT%  ORB  DRB  TRB   \n",
       "101  Rk  Player  Pos  Age  Tm  G  GS  MP  FG  FGA  ...  FT%  ORB  DRB  TRB   \n",
       "130  Rk  Player  Pos  Age  Tm  G  GS  MP  FG  FGA  ...  FT%  ORB  DRB  TRB   \n",
       "\n",
       "     AST  STL  BLK  TOV  PF  PTS  \n",
       "22   AST  STL  BLK  TOV  PF  PTS  \n",
       "53   AST  STL  BLK  TOV  PF  PTS  \n",
       "76   AST  STL  BLK  TOV  PF  PTS  \n",
       "101  AST  STL  BLK  TOV  PF  PTS  \n",
       "130  AST  STL  BLK  TOV  PF  PTS  \n",
       "\n",
       "[5 rows x 30 columns]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Assigns the table in a variable df_2020\n",
    "\n",
    "df_2020 = df[0]\n",
    "\n",
    "# Let check the table header which is presented multiple times in several rows\n",
    "\n",
    "df_2020[df_2020.Age == 'Age'].head() #  All the subsequent table header selected for this entire dataframe!\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " total numbers of redundant headers: 26 \n",
      " total rows of df_2020:     677 \n",
      " total rows of df_2020_new: 651 \n",
      "===========================================\n"
     ]
    },
    {
     "data": {
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       "<div>\n",
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       "    <tr style=\"text-align: right;\">\n",
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       "      <th>Rk</th>\n",
       "      <th>Player</th>\n",
       "      <th>Pos</th>\n",
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       "      <th>DRB</th>\n",
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       "      <th>STL</th>\n",
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       "      <th>TOV</th>\n",
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       "      <th>0</th>\n",
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       "      <td>26</td>\n",
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       "      <td>63</td>\n",
       "      <td>63</td>\n",
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       "      <td>.582</td>\n",
       "      <td>3.3</td>\n",
       "      <td>6.0</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>Bam Adebayo</td>\n",
       "      <td>PF</td>\n",
       "      <td>22</td>\n",
       "      <td>MIA</td>\n",
       "      <td>72</td>\n",
       "      <td>72</td>\n",
       "      <td>33.6</td>\n",
       "      <td>6.1</td>\n",
       "      <td>11.0</td>\n",
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       "      <td>.691</td>\n",
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       "      <td>7.8</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>LaMarcus Aldridge</td>\n",
       "      <td>C</td>\n",
       "      <td>34</td>\n",
       "      <td>SAS</td>\n",
       "      <td>53</td>\n",
       "      <td>53</td>\n",
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       "      <td>7.4</td>\n",
       "      <td>15.0</td>\n",
       "      <td>...</td>\n",
       "      <td>.827</td>\n",
       "      <td>1.9</td>\n",
       "      <td>5.5</td>\n",
       "      <td>7.4</td>\n",
       "      <td>2.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>2.4</td>\n",
       "      <td>18.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>Kyle Alexander</td>\n",
       "      <td>C</td>\n",
       "      <td>23</td>\n",
       "      <td>MIA</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>6.5</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.0</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.0</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>Nickeil Alexander-Walker</td>\n",
       "      <td>SG</td>\n",
       "      <td>21</td>\n",
       "      <td>NOP</td>\n",
       "      <td>47</td>\n",
       "      <td>1</td>\n",
       "      <td>12.6</td>\n",
       "      <td>2.1</td>\n",
       "      <td>5.7</td>\n",
       "      <td>...</td>\n",
       "      <td>.676</td>\n",
       "      <td>0.2</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.8</td>\n",
       "      <td>1.9</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>1.1</td>\n",
       "      <td>1.2</td>\n",
       "      <td>5.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "      <td>Grayson Allen</td>\n",
       "      <td>SG</td>\n",
       "      <td>24</td>\n",
       "      <td>MEM</td>\n",
       "      <td>38</td>\n",
       "      <td>0</td>\n",
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       "      <td>3.1</td>\n",
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       "      <td>.867</td>\n",
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       "      <td>2.0</td>\n",
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       "      <td>1.4</td>\n",
       "      <td>0.3</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.9</td>\n",
       "      <td>1.4</td>\n",
       "      <td>8.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>7</td>\n",
       "      <td>Jarrett Allen</td>\n",
       "      <td>C</td>\n",
       "      <td>21</td>\n",
       "      <td>BRK</td>\n",
       "      <td>70</td>\n",
       "      <td>64</td>\n",
       "      <td>26.5</td>\n",
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       "      <td>11.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8</td>\n",
       "      <td>Kadeem Allen</td>\n",
       "      <td>PG</td>\n",
       "      <td>27</td>\n",
       "      <td>NYK</td>\n",
       "      <td>10</td>\n",
       "      <td>0</td>\n",
       "      <td>11.7</td>\n",
       "      <td>1.9</td>\n",
       "      <td>4.4</td>\n",
       "      <td>...</td>\n",
       "      <td>.636</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.7</td>\n",
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       "      <td>2.1</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.7</td>\n",
       "      <td>5.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>9</td>\n",
       "      <td>Al-Farouq Aminu</td>\n",
       "      <td>PF</td>\n",
       "      <td>29</td>\n",
       "      <td>ORL</td>\n",
       "      <td>18</td>\n",
       "      <td>2</td>\n",
       "      <td>21.1</td>\n",
       "      <td>1.4</td>\n",
       "      <td>4.8</td>\n",
       "      <td>...</td>\n",
       "      <td>.655</td>\n",
       "      <td>1.3</td>\n",
       "      <td>3.5</td>\n",
       "      <td>4.8</td>\n",
       "      <td>1.2</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.9</td>\n",
       "      <td>1.5</td>\n",
       "      <td>4.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>10</td>\n",
       "      <td>Justin Anderson</td>\n",
       "      <td>SG</td>\n",
       "      <td>26</td>\n",
       "      <td>BRK</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "      <td>10.7</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.8</td>\n",
       "      <td>...</td>\n",
       "      <td>.500</td>\n",
       "      <td>0.1</td>\n",
       "      <td>2.0</td>\n",
       "      <td>2.1</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.4</td>\n",
       "      <td>1.3</td>\n",
       "      <td>2.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>11</td>\n",
       "      <td>Kyle Anderson</td>\n",
       "      <td>SF</td>\n",
       "      <td>26</td>\n",
       "      <td>MEM</td>\n",
       "      <td>67</td>\n",
       "      <td>28</td>\n",
       "      <td>19.9</td>\n",
       "      <td>2.3</td>\n",
       "      <td>4.9</td>\n",
       "      <td>...</td>\n",
       "      <td>.667</td>\n",
       "      <td>0.9</td>\n",
       "      <td>3.4</td>\n",
       "      <td>4.3</td>\n",
       "      <td>2.4</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.7</td>\n",
       "      <td>5.8</td>\n",
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       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>12</td>\n",
       "      <td>Ryan Anderson</td>\n",
       "      <td>C</td>\n",
       "      <td>31</td>\n",
       "      <td>HOU</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.5</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>3.5</td>\n",
       "      <td>3.5</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.5</td>\n",
       "      <td>2.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>13</td>\n",
       "      <td>Giannis Antetokounmpo</td>\n",
       "      <td>PF</td>\n",
       "      <td>25</td>\n",
       "      <td>MIL</td>\n",
       "      <td>63</td>\n",
       "      <td>63</td>\n",
       "      <td>30.4</td>\n",
       "      <td>10.9</td>\n",
       "      <td>19.7</td>\n",
       "      <td>...</td>\n",
       "      <td>.633</td>\n",
       "      <td>2.2</td>\n",
       "      <td>11.4</td>\n",
       "      <td>13.6</td>\n",
       "      <td>5.6</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.7</td>\n",
       "      <td>3.1</td>\n",
       "      <td>29.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>14</td>\n",
       "      <td>Kostas Antetokounmpo</td>\n",
       "      <td>PF</td>\n",
       "      <td>22</td>\n",
       "      <td>LAL</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.6</td>\n",
       "      <td>...</td>\n",
       "      <td>.500</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.4</td>\n",
       "      <td>1.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>15</td>\n",
       "      <td>Thanasis Antetokounmpo</td>\n",
       "      <td>SF</td>\n",
       "      <td>27</td>\n",
       "      <td>MIL</td>\n",
       "      <td>20</td>\n",
       "      <td>2</td>\n",
       "      <td>6.5</td>\n",
       "      <td>1.2</td>\n",
       "      <td>2.4</td>\n",
       "      <td>...</td>\n",
       "      <td>.412</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.2</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.4</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.6</td>\n",
       "      <td>0.9</td>\n",
       "      <td>2.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>16</td>\n",
       "      <td>Carmelo Anthony</td>\n",
       "      <td>PF</td>\n",
       "      <td>35</td>\n",
       "      <td>POR</td>\n",
       "      <td>58</td>\n",
       "      <td>58</td>\n",
       "      <td>32.8</td>\n",
       "      <td>5.8</td>\n",
       "      <td>13.5</td>\n",
       "      <td>...</td>\n",
       "      <td>.845</td>\n",
       "      <td>1.2</td>\n",
       "      <td>5.1</td>\n",
       "      <td>6.3</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.8</td>\n",
       "      <td>0.5</td>\n",
       "      <td>1.7</td>\n",
       "      <td>2.9</td>\n",
       "      <td>15.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>17</td>\n",
       "      <td>OG Anunoby</td>\n",
       "      <td>SF</td>\n",
       "      <td>22</td>\n",
       "      <td>TOR</td>\n",
       "      <td>69</td>\n",
       "      <td>68</td>\n",
       "      <td>29.9</td>\n",
       "      <td>4.1</td>\n",
       "      <td>8.2</td>\n",
       "      <td>...</td>\n",
       "      <td>.706</td>\n",
       "      <td>1.2</td>\n",
       "      <td>4.1</td>\n",
       "      <td>5.3</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.4</td>\n",
       "      <td>0.7</td>\n",
       "      <td>1.1</td>\n",
       "      <td>2.4</td>\n",
       "      <td>10.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>18</td>\n",
       "      <td>Ryan Arcidiacono</td>\n",
       "      <td>PG</td>\n",
       "      <td>25</td>\n",
       "      <td>CHI</td>\n",
       "      <td>58</td>\n",
       "      <td>4</td>\n",
       "      <td>16.0</td>\n",
       "      <td>1.6</td>\n",
       "      <td>3.8</td>\n",
       "      <td>...</td>\n",
       "      <td>.711</td>\n",
       "      <td>0.3</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.9</td>\n",
       "      <td>1.7</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.6</td>\n",
       "      <td>1.7</td>\n",
       "      <td>4.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>19</td>\n",
       "      <td>Trevor Ariza</td>\n",
       "      <td>SF</td>\n",
       "      <td>34</td>\n",
       "      <td>TOT</td>\n",
       "      <td>53</td>\n",
       "      <td>21</td>\n",
       "      <td>28.2</td>\n",
       "      <td>2.7</td>\n",
       "      <td>6.1</td>\n",
       "      <td>...</td>\n",
       "      <td>.838</td>\n",
       "      <td>0.6</td>\n",
       "      <td>4.0</td>\n",
       "      <td>4.6</td>\n",
       "      <td>1.7</td>\n",
       "      <td>1.3</td>\n",
       "      <td>0.3</td>\n",
       "      <td>1.1</td>\n",
       "      <td>2.1</td>\n",
       "      <td>8.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>19</td>\n",
       "      <td>Trevor Ariza</td>\n",
       "      <td>SF</td>\n",
       "      <td>34</td>\n",
       "      <td>SAC</td>\n",
       "      <td>32</td>\n",
       "      <td>0</td>\n",
       "      <td>24.7</td>\n",
       "      <td>2.0</td>\n",
       "      <td>5.2</td>\n",
       "      <td>...</td>\n",
       "      <td>.778</td>\n",
       "      <td>0.7</td>\n",
       "      <td>3.9</td>\n",
       "      <td>4.6</td>\n",
       "      <td>1.6</td>\n",
       "      <td>1.1</td>\n",
       "      <td>0.2</td>\n",
       "      <td>0.9</td>\n",
       "      <td>2.0</td>\n",
       "      <td>6.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>20 rows × 30 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    Rk                    Player Pos Age   Tm   G  GS    MP    FG   FGA  ...  \\\n",
       "0    1              Steven Adams   C  26  OKC  63  63  26.7   4.5   7.6  ...   \n",
       "1    2               Bam Adebayo  PF  22  MIA  72  72  33.6   6.1  11.0  ...   \n",
       "2    3         LaMarcus Aldridge   C  34  SAS  53  53  33.1   7.4  15.0  ...   \n",
       "3    4            Kyle Alexander   C  23  MIA   2   0   6.5   0.5   1.0  ...   \n",
       "4    5  Nickeil Alexander-Walker  SG  21  NOP  47   1  12.6   2.1   5.7  ...   \n",
       "5    6             Grayson Allen  SG  24  MEM  38   0  18.9   3.1   6.6  ...   \n",
       "6    7             Jarrett Allen   C  21  BRK  70  64  26.5   4.3   6.6  ...   \n",
       "7    8              Kadeem Allen  PG  27  NYK  10   0  11.7   1.9   4.4  ...   \n",
       "8    9           Al-Farouq Aminu  PF  29  ORL  18   2  21.1   1.4   4.8  ...   \n",
       "9   10           Justin Anderson  SG  26  BRK  10   1  10.7   1.0   3.8  ...   \n",
       "10  11             Kyle Anderson  SF  26  MEM  67  28  19.9   2.3   4.9  ...   \n",
       "11  12             Ryan Anderson   C  31  HOU   2   0   7.0   1.0   3.5  ...   \n",
       "12  13     Giannis Antetokounmpo  PF  25  MIL  63  63  30.4  10.9  19.7  ...   \n",
       "13  14      Kostas Antetokounmpo  PF  22  LAL   5   0   4.0   0.6   0.6  ...   \n",
       "14  15    Thanasis Antetokounmpo  SF  27  MIL  20   2   6.5   1.2   2.4  ...   \n",
       "15  16           Carmelo Anthony  PF  35  POR  58  58  32.8   5.8  13.5  ...   \n",
       "16  17                OG Anunoby  SF  22  TOR  69  68  29.9   4.1   8.2  ...   \n",
       "17  18          Ryan Arcidiacono  PG  25  CHI  58   4  16.0   1.6   3.8  ...   \n",
       "18  19              Trevor Ariza  SF  34  TOT  53  21  28.2   2.7   6.1  ...   \n",
       "19  19              Trevor Ariza  SF  34  SAC  32   0  24.7   2.0   5.2  ...   \n",
       "\n",
       "     FT%  ORB   DRB   TRB  AST  STL  BLK  TOV   PF   PTS  \n",
       "0   .582  3.3   6.0   9.3  2.3  0.8  1.1  1.5  1.9  10.9  \n",
       "1   .691  2.4   7.8  10.2  5.1  1.1  1.3  2.8  2.5  15.9  \n",
       "2   .827  1.9   5.5   7.4  2.4  0.7  1.6  1.4  2.4  18.9  \n",
       "3    NaN  1.0   0.5   1.5  0.0  0.0  0.0  0.5  0.5   1.0  \n",
       "4   .676  0.2   1.6   1.8  1.9  0.4  0.2  1.1  1.2   5.7  \n",
       "5   .867  0.2   2.0   2.2  1.4  0.3  0.1  0.9  1.4   8.7  \n",
       "6   .633  3.1   6.5   9.6  1.6  0.6  1.3  1.1  2.3  11.1  \n",
       "7   .636  0.2   0.7   0.9  2.1  0.5  0.2  0.8  0.7   5.0  \n",
       "8   .655  1.3   3.5   4.8  1.2  1.0  0.4  0.9  1.5   4.3  \n",
       "9   .500  0.1   2.0   2.1  0.8  0.0  0.6  0.4  1.3   2.8  \n",
       "10  .667  0.9   3.4   4.3  2.4  0.8  0.6  1.0  1.7   5.8  \n",
       "11   NaN  0.0   3.5   3.5  1.0  0.5  0.0  0.5  0.5   2.5  \n",
       "12  .633  2.2  11.4  13.6  5.6  1.0  1.0  3.7  3.1  29.5  \n",
       "13  .500  0.4   0.2   0.6  0.4  0.0  0.0  0.2  0.4   1.4  \n",
       "14  .412  0.6   0.6   1.2  0.8  0.4  0.1  0.6  0.9   2.8  \n",
       "15  .845  1.2   5.1   6.3  1.5  0.8  0.5  1.7  2.9  15.4  \n",
       "16  .706  1.2   4.1   5.3  1.6  1.4  0.7  1.1  2.4  10.6  \n",
       "17  .711  0.3   1.6   1.9  1.7  0.5  0.1  0.6  1.7   4.5  \n",
       "18  .838  0.6   4.0   4.6  1.7  1.3  0.3  1.1  2.1   8.0  \n",
       "19  .778  0.7   3.9   4.6  1.6  1.1  0.2  0.9  2.0   6.0  \n",
       "\n",
       "[20 rows x 30 columns]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Checking the length for how many header we have:\n",
    "\n",
    "print(f' total numbers of redundant headers: {len(df_2020[df_2020.Age == \"Age\"])} ')\n",
    "\n",
    "# Drop the redundant headers in the dataframe:\n",
    "df_2020_new = df_2020.drop(df_2020[df_2020.Age == 'Age'].index)\n",
    "\n",
    "# Compare before and after dropping redundant headers with numbers of rows:\n",
    "\n",
    "print(f' total rows of df_2020:     {df_2020.shape[0]} ')\n",
    "print(f' total rows of df_2020_new: {df_2020_new.shape[0]} ')\n",
    "print('===========================================')\n",
    "\n",
    "df_2020_new.head(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quick Exploratory Data Analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Making a simple histogram\n",
    "\n",
    "plt.figure(figsize=(10,8))\n",
    "\n",
    "sns.distplot(df_2020_new.PTS,    # Checking frequency of the player points\n",
    "            kde= False,          # Should be False because we want to retain the original frequency ( \"kde=True\" => it will be the probability)\n",
    "            hist_kws = dict( edgecolor = 'black', linewidth=2))  \n",
    "\n",
    "plt.title('HISTOGRAM OF PLAYER POINTS PER GAME IN THE 2020 NBA SEASON')\n",
    "plt.ylabel('NUMBERS OF PLAYERS')\n",
    "plt.xlabel('POINTS PER GAME')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the histogram, we can see: \n",
    "- There are about 57 players having between 0 and 1 point.\n",
    "- There are less than 10 players who had 30 points."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}