__000__Terminale Proba Python Evenement_ensemble.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chargement de quelques modules:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "print('venn3' in sys.modules)\n",
    "print('matplotlib' in sys.modules)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "id": "51RjKvWIrree"
   },
   "outputs": [],
   "source": [
    "#from matplotlib_venn import venn3\n",
    "from matplotlib import pyplot as plt\n",
    "from fractions import Fraction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "YQczJFaMv4EP"
   },
   "source": [
    "# Définissons trois ensembles:\n",
    "\n",
    "  * Un ensemble $A$ comprenant 7 cartes: $A = \\{roi,dame,sept,as,dix,six,valet\\}$\n",
    "  * Un ensemble $B$ comprenant 3 cartes: $B = \\{dame,deux,dix\\}$\n",
    "\n",
    "ainsi que l'ensemble $U$ réunion des ensembles $A$, $B$ et $\\{trois, quatre, cinq\\}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "id": "fJhTCn2-yBdX"
   },
   "outputs": [],
   "source": [
    "A = {'roi','dame','sept','as','dix','six','valet'}\n",
    "B = {'dame','deux','dix'}\n",
    "C = {'trois','quatre','cinq'}\n",
    "U = A | B | C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "SHehlqCbyWUk"
   },
   "source": [
    "Vérifions le type de $U$ : c'est un ensemble (set)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "_dV3Td-LrPCV",
    "outputId": "999e5246-ebc3-44d5-eda7-e85f334d1a68"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "set"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "C-V0pOnWyf8P"
   },
   "source": [
    "Quels sont les éléments de $U$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 219
    },
    "id": "J7nOkDhMrPCo",
    "outputId": "2d7ebeeb-a7c6-4f8a-e77b-109499e22837"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'as',\n",
       " 'cinq',\n",
       " 'dame',\n",
       " 'deux',\n",
       " 'dix',\n",
       " 'quatre',\n",
       " 'roi',\n",
       " 'sept',\n",
       " 'six',\n",
       " 'trois',\n",
       " 'valet'}"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "GDZD_p5Q0ni1"
   },
   "source": [
    "## Représentons les ensembles avec un diagramme de Venn:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 338
    },
    "id": "H5x0Ouj5r7RM",
    "outputId": "3c7dec24-fda5-4258-8290-8d19303dfda1"
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'venn3' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-16-e5748668f932>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m121\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mvenn3\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mB\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m'A'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'B'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'C'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      5\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      6\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m122\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'venn3' is not defined"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x1152 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(12,16))\n",
    "\n",
    "plt.subplot(121)\n",
    "venn3([A, B, C], ('A', 'B', 'C'))\n",
    "\n",
    "plt.subplot(122)\n",
    "venn3([A, B, U], ('A', 'B', 'U'))\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "lpKjEcT700X4"
   },
   "source": [
    "## Opération sur les ensembles:\n",
    "\n",
    "  * Sous ensemble\n",
    "  * Intersection de deux ensembles\n",
    "  * Union de deux ensembles\n",
    "  * nombre d'éléments d'un ensemble"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6N9fknMb1aOT"
   },
   "source": [
    "### Sous ensemble:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 54
    },
    "id": "D6G-RPJVrPC4",
    "outputId": "0463e9d6-5e0c-4d45-ba7b-168107937693"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "False\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "print(B.issubset(A))\n",
    "print(A.issubset(U))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 72
    },
    "id": "Am98GQoG6fDW",
    "outputId": "cb33e4b8-ddf4-45bb-bac6-6e6ea931b497"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "True\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "print(U <= U)\n",
    "print(U >=U)\n",
    "print(U >= A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "5-0vHgV21gO1"
   },
   "source": [
    "###intersection de deux ensembles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "uOFggf0arPDF"
   },
   "outputs": [],
   "source": [
    "A_inter_B = B.intersection(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "_jtW-n1IrPDO",
    "outputId": "beb427f8-4932-4d7a-e4f9-068a22b639e8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'dame', 'dix'}"
      ]
     },
     "execution_count": 9,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_inter_B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "D9O2zuTBrPDc",
    "outputId": "07c5b32f-e06a-4600-def7-36dcf4c5c57f"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'dame', 'dix'}"
      ]
     },
     "execution_count": 10,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A & B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "QMrLM6bHrPDp",
    "outputId": "003e88d7-e73a-4c71-a411-40cbc4fc3fd9"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'dame', 'dix'}"
      ]
     },
     "execution_count": 11,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B & A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "aTGm6e_F1mAh"
   },
   "source": [
    "### Union de deux ensembles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "UHLJZfmdrPD3",
    "outputId": "940d40eb-81ac-4400-a407-659a76a759c3"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'as', 'dame', 'deux', 'dix', 'roi', 'sept', 'six', 'valet'}"
      ]
     },
     "execution_count": 12,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A | B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "10EbJy9a1wfk"
   },
   "source": [
    "### Nombre d'éléments d'un ensemble:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "J2DQffpRrPEE",
    "outputId": "339e6ba1-66ca-446a-aa77-bafe74526fa8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 13,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(A | B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ugV41RF627_N"
   },
   "source": [
    "# Probabilité d'un événement:\n",
    "\n",
    "   * Probabilité de l'événement $A$ : $P(A) = \\frac{Nombre~de~cas~favorables}\n",
    "   {Nombre~de~cas~possibles}$\n",
    "   * Proba de A et B : $P(A \\cap B)$\n",
    "   * Proba de A ou B : $P(A \\cup B)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uB4p7ntprPER"
   },
   "source": [
    " **Calculons une proba en python:**\n",
    "\n",
    " Avec une valeur approchée (argh!!)\n",
    "``` python\n",
    "    pA= len(A)/len(U)\n",
    "```\n",
    "maintenant avec une valeur exacte et le module *Fraction* importé plus haut:\n",
    "``` python\n",
    "    pA= Fraction(len(A),len(U))\n",
    "```\n",
    "On respire mieux. Profitons-en pour vérifier si on a bien :\n",
    "\n",
    "$$P(A \\cup B) = P(A)+P(B)-P(A  \\cap B)$$\n",
    "\n",
    "en montrant que:\n",
    "\n",
    "```python\n",
    "    pA_ou_B == pA+pB-pAB\n",
    "```\n",
    "est vrai"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 109
    },
    "id": "t1eucJey4AeI",
    "outputId": "fed01244-87d7-4f48-d616-70e66c23713e"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "proba de A: 0.6363636363636364\n",
      "proba de A avec fraction: 7/11\n",
      "proba de B avec fraction: 3/11\n",
      "proba de A et B,  avec fraction: 2/11\n",
      "proba de A ou B,  avec fraction: 8/11\n"
     ]
    }
   ],
   "source": [
    "print('proba de A:', len(A)/len(U))\n",
    "pA = Fraction(len(A),len(U))\n",
    "print('proba de A avec fraction:', pA)\n",
    "pB = Fraction(len(B),len(U))\n",
    "pAB = Fraction(len(A & B),len(U))\n",
    "pA_ou_B = Fraction(len(A | B), len(U))\n",
    "print('proba de B avec fraction:', pB)\n",
    "\n",
    "print('proba de A et B,  avec fraction:', pAB)\n",
    "print('proba de A ou B,  avec fraction:', pA_ou_B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 35
    },
    "id": "qZADggEhrPEW",
    "outputId": "f1b54d27-58fb-4b49-c72b-f4af0361d067"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 38,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pA_ou_B == pA+pB-pAB"
   ]
  }
 ],
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   "name": "Evenement_ensemble.ipynb",
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   "language": "python",
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