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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Al\u00e9atoire : une Introduction aux Probabilit\u00e9s (semaine 1)\n",
      "\n",
      "J'ai cr\u00e9\u00e9 ce \u00ab notebook \u00bb avec [iPython Notebook](http://ipython.org/notebook.html), un syst\u00e8me gratuit (et \u00ab libre \u00bb) qui peut faire des calculs symboliques, cr\u00e9er des graphiques, afficher des expressions math\u00e9matiques avec $\\LaTeX$ et faire beaucoup d'autres choses utiles.\n",
      "\n",
      "On commence par configurer plusieurs biblioth\u00e8ques qu'on va utiliser bient\u00f4t."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.interactive import printing\n",
      "printing.init_printing()\n",
      "import sympy as sym\n",
      "from sympy.mpmath import limit, inf\n",
      "from sympy import Rational as frac\n",
      "%pylab inline --no-import-all"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercice 1 : Le \u00ab paradoxe \u00bb du chevalier de M\u00e9r\u00e9\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\Omega_1 & = \\{1,2,3,4,5,6\\}^4 \\\\\n",
      "\\mathbb{P}_1(\\omega_i) & = \\left(\\frac{1}{6}\\right)^4 \\\\\n",
      "\\mathbb{P}_1(A_1) & = 1-\\left(\\frac{5}{6}\\right)^4\n",
      "\\end{aligned}$$\n",
      "\n",
      "On calcule la probabilit\u00e9 de $A_1^C$, la probabilit\u00e9 qu'aucun 6 appara\u00eet dans 4 lancers d'un d\u00e9 \u00e0 6 faces, est on utilise \u00e7a pour calculer $\\mathbb{P}(A_1) = 1-\\mathbb{P}(A_1^C)$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1-frac(5,6)**4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{671}{1296}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAArBAMAAAD8sQfNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nABcbhU70Ib4pAAAAAElFTkSuQmCC\n",
       "prompt_number": 2,
       "text": [
        "671 \n",
        "\u2500\u2500\u2500\u2500\n",
        "1296"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1-(5.0/6)**4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0.51774691358$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAPBAMAAADe9tr1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAACVUlEQVQoFaWST2sTQRjGf7ubmGaTJosWESokWvQiLcFWEP/QULVXQwVvQqqCl6r9\nAtLevLQQ9CZCF+mth7YXiyA2ehFRNId6aS0N4heoStK/xmem2/oBXNhnZt555sf7zjtwuO8U5rsb\nDNXhUEB8squ314qiQ/q9vnM4XQMVTYf1P+19H7k7rvRofY9jZovF1m9IfgnwW63WrhVFJ8B5wGk6\nw+QreL4uS56xcM+dp30Ufw6vLCd3lio479YCXPFqViD2WkkV+MQKfON4twCxJtlx606EJMukayQU\n1hkjXA/wlHTJCjwbgVmTYAP6A2JyZt6QK1u3V8UZJ1sj0zRHDwCa/zABIwUBPmp0dsSuW4BWKsG4\nMzuBF5LLk1HxsDxtLlMZ6KtG4lcEaNwYqPNSGYT7gNuRe23rAkwViP8y/rdM1SOAuCpD0sEITiNk\ngX54WNwDxM/nI7fbUlcOAJAejwA5ncdIwQBaAWeCQ8XEvwwulrQn94ub23VyhagEcDcjwKABSBIl\nAdjQPdaZvtY/ul9Cal777qZfY3aebJ6EucT2Mt6fCDBhAJIjGIDKnw0VUIdNF5IV0k3rTunRNWmr\n4ps2psq4Gs0lOrtaG3m0urq9ZLqgDOC+3oAc2XXSW9ZtilwmNodb1syFNo0GEP+ptRXQQxrTHTBT\niatAA0hVadu07lQFPsBjOktOw7zQGQUMIGEAVkD9TxecBQaDo+EewA8ZK1p3Zhi/qEKvfoVJeHL5\nBHQvTup9LghghUutZTjbXcI7+VlZrmxo+d1MrftWV4+c//n9BZkrxSliKH75AAAAAElFTkSuQmCC\n",
       "prompt_number": 3,
       "text": [
        "0.51774691358"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\\begin{aligned}\n",
      "\\Omega_2 & = (\\{1,2,3,4,5,6\\} \\times \\{1,2,3,4,5,6\\})^{24} \\\\\n",
      "\\mathbb{P}_2(\\omega_i) & = \\left(\\frac{1}{36}\\right)^{24} \\\\\n",
      "\\mathbb{P}_2(A_2) &= 1 - \\left(\\frac{35}{36}\\right)^{24}\n",
      "\\end{aligned}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1-frac(35,36)**24"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{11033126465283976852912127963392284191}{22452257707354557240087211123792674816}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYsAAAArBAMAAAB7pbniAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nsDVRZC3KUkaqCXyZtv8Hx38AJcKO4lRjDvMAAAAASUVORK5CYII=\n",
       "prompt_number": 4,
       "text": [
        "11033126465283976852912127963392284191\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "22452257707354557240087211123792674816"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1-(35.0/36)**24"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0.491403876131$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAPBAMAAAAmFiECAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAACgUlEQVQ4EaWSTWgTQRiGn8kmu82mSRfbg6CQYFGQ2rK01YP0EGrxUsRgwZs0BUGQ\nIAviUSpeRGgxePMHDB4E9ZB6qKJWXLyJSnLQSyU0B28e/Cu19Mf4zewqeHYuO/O97zz7/QzsGN1L\ntE4gh/2ytz0dSBVQ/eMl0iNjAfQdGSI13z8yAtboQZFniJXIfoadVX0L5lDPuFUl3TCYrhA7UFe5\njrMBBbKB2+l0tlHn2Ae3vxErxu4uYJWFAcklsgtkQ/VqxWAmQx7CIY4HrOPUSJcT8r8mts9bdg8K\nxiiRvbuJIxFZNyv0lLG0rDHpeyHvYNb73FZrWCHqkiUFlagLTH4qPqNE9p4muZ86jl8hH5L8FWMs\nO2QbvgToonJbnlUT0yd4I58IY5TIni+QW9Vxt1ohI9lIH0w29+1QfRdMUVpegJWNw9oVwtrUePsP\nRivGfscn9UP29FHB+ootTI1RRTtMizDtM3VZ5ERHRoNkpNZqPIkxRvkX4wuG05yPs7H4i8FahLsn\nN9uQF37HY8AzvRGsKAaT96OinJLGuBNT0iidzQXBKMlGF8Vzz21SfwRH5bAO9XaMESWy9xRwdIt7\n0RjIRpNSPnaI9GY2uAGNIOOREtucOB4LpmYwRokw8shcucnFVmvzvXztpok7rdbKi6LcqntSRKMm\n1bCMktHpScXZGCXCJBdIlEWUtYT7lOlqFIdMyCTyYJfgWjUj4dekJD15SQxELTZKbL/CrpK8L1lb\nJGfUWdmYgdMT0hWoRR7grJKbwS3iaEy3r+JJGSW29058gHmRxzrLDB9ow+DL+ZoMobFdU6PHSuT2\n6OCp/iF52XIfhgdLJD6uL8dKZNfCf6/fKijVpHsKCGwAAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        "0.491403876131"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On peut voir que $\\mathbb{P}_1(A_1)$ est superieur \u00e0 $\\mathbb{P}_2(A_2)$, mais par une petite quantit\u00e9."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def p_a(n):\n",
      "    denom = 6**n\n",
      "    num = denom - 1\n",
      "    return 1 - (num/denom)**(2*denom/3)\n",
      "\n",
      "x = np.linspace(1,6)\n",
      "plt.plot(x, p_a(x))\n",
      "None"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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UYyP84Q/w1ltG19HKSuME8P3vw/Tp8N3v6lkCkStByV9M9cUXsG+fcSJ45x04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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4fc3610>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La limite de $\\mathbb{P}_n(A_n)$ lorsque $n$ tend vers l'infini a l'air d'\u00eatre un peu moins que 0.49.\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\lim_{n \\to \\infty} \\left[ 1 - \\left(\\frac{6^n-1}{6^n}\\right)^{\\frac{2}{3} n} \\right]\n",
      "\\end{aligned}$$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(p_a, inf)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "mpf('0.48658288096740798')"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercice 2 : Jeu du Blackjack\n",
      "\n",
      "$$\n",
      "\\Omega = \\{ 2,3,4,5,6,7,8,9,10, \\text{valet }(10), \\text{dame }(10), \\text{roi }(10), \\text{as }(1\\text{ ou }11) \\} \\times \\{ \\heartsuit, \\diamondsuit, \\clubsuit, \\spadesuit \\}\n",
      "$$\n",
      "\n",
      "On pioche deux cartes, sans replacement. On a trois cas : Deux as, un as et une autre carte, et deux non-as.\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(\\text{deux as}) &= \\frac{4}{52}\\frac{3}{51}\\\\\n",
      "\\mathbb{P}(\\text{as et un non-as}) &= \\frac{4}{52}\\frac{48}{51} + \\frac{48}{52}\\frac{4}{51} = 2\\left(\\frac{4}{52}\\frac{48}{51}\\right)\\\\\n",
      "\\mathbb{P}(\\text{deux non-as}) &= \\frac{48}{52}\\frac{47}{51}\n",
      "\\end{aligned}$$\n",
      "\n",
      "Si on pioche deux as, $\\mathbb{P}(\\text{blackjack} | \\text{deux as}) = 0$, car $1+1=2$, $1+11=12$ et $11+11=22$. Si on pioche un as, l'autre carte doit avoir une valeur de 10, et donc $\\mathbb{P}(\\text{blackjack} | \\text{un as et un non-as}) = \\frac{4}{12} = \\frac{1}{3}$. Et sans au moins un as, on ne peut pas ganger, donc on a $\\mathbb{P}(\\text{blackjack} | \\text{deux non-as}) = 0$.\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(\\text{blackjack}) = 0 + \\frac{2}{3}\\left(\\frac{4}{52}\\frac{48}{51}\\right) + 0\n",
      "\\end{aligned}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "frac(2,3)*frac(4,52)*frac(48,51)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{32}{663}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAqBAMAAAAgz0ShAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAABN0lEQVQoFc2RsUoDQRBA3+WSS+40JAqC9YkoVkFtrHKNdQ7CFXaprMUf0MbaNDZW\nARtNEFKLmCBoJwl+gaUgIkqICRHO2TUbxV5wYd/OPmZ2BhbA8hcCrGitK7FecyT7XJEYGrEXMOAJ\n6kYcV6c+OIWXSY0q6TR/iulQZR82Tc39gYrcN3OHxLnEmcK3oC7Z5cl9BooB6YeJiJsUK2xi5cfq\nRBrksyFpI7ZJvjPvLz6aGjcqV+nE8cCIf3HGv9ZfDeXvdkFh1l9XPS6rqbaGFVKqiLjGLmike+T2\nwXkVqeGe0WqDXROhIacqyd0tbX1BxI3s1i1eoEF2OVRiRKqhIZcVGUAedoYaIryG7DZOX0N+0u7p\nBs5QukjaCLsPSXmjpuEVyKihVrkINJwKpR0Rieh5jKNoAz4B7aFtS/68x4wAAAAASUVORK5CYII=\n",
       "prompt_number": 8,
       "text": [
        " 32\n",
        "\u2500\u2500\u2500\n",
        "663"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si le joueur a une dame de $\\spadesuit$ et un 5 de $\\clubsuit$, il faut qu'il re\u00e7oive une carte avec une valeur de 1, 2, 3, 4, 5, 6 ou 7 pour ne pas perdre. Il reste $4 \\cdot 7 - 1 = 27$ cartes sur 50 avec la valeur necessaire, et donc il a 27 chances sur 50 de ne pas perdre."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercice 3 : La formule d\u2019inclusion-exclusion\n",
      "\n",
      "On veut d\u00e9montrer que :\n",
      "\n",
      "$$\n",
      "\\mathbb{P}(\\cup_{m=1}^{n} A_m) = \\sum_{k=1}^n (-1)^{k+1} p_k\n",
      "$$\n",
      "\n",
      "ou\n",
      "\n",
      "$$\n",
      "p_k = \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq n} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_k}).\n",
      "$$\n",
      "\n",
      "Quand $n$ est \u00e9gal \u00e0 1, on a\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(A_1) &= (-1)^2 \\mathbb{P}(A_1) \\\\\n",
      "                &= \\mathbb{P}(A_1).\n",
      "\\end{aligned}$$\n",
      "\n",
      "Quand $n$ est \u00e9gal \u00e0 2, on a\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(A_1 \\cup A_2) &= (-1)^2 (\\mathbb{P}(A_1) + \\mathbb{P}(A_2)) + (-1)^3(\\mathbb{P}(A_1 \\cap A_2)) \\\\\n",
      "                         &= \\mathbb{P}(A_1) + \\mathbb{P}(A_2) - \\mathbb{P}(A_1 \\cap A_2).\n",
      "\\end{aligned}$$\n",
      "\n",
      "Quand on ajoute $\\mathbb{P}(A_1)$ \u00e0 $\\mathbb{P}(A_2)$, on compte $\\mathbb{P}(A_1 \\cap A_2)$ deux fois, et il faut donc le soustraire.\n",
      "\n",
      "Soit $B$ \u00e9gal \u00e0 $\\cup_{m=1}^{n} A_m$. On peut voir que\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(B \\cup A_{n+1}) &= \\mathbb{P}(B) + \\mathbb{P}(A_{n+1}) - \\mathbb{P}(B \\cap A_{n+1}) \\\\\n",
      "  &= \\mathbb{P}(B) + \\mathbb{P}(A_{n+1}) - \\mathbb{P}(\\cup_{m=1}^{n} (A_m \\cap A_{n+1})) \\\\\n",
      "  &= \\left[\\sum_{k=1}^n (-1)^{k+1} p_k\\right] + \\mathbb{P}(A_{n+1}) - \\left[\\sum_{k=1}^n (-1)^{k+1} q_k\\right]\n",
      "\\end{aligned}$$\n",
      "\n",
      "ou\n",
      "\n",
      "$$\\begin{aligned}\n",
      "p_k &= \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq n} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_k}) & \\text{et}\\\\\n",
      "q_k &= \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq n} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_k} \\cap A_{n+1}).\n",
      "\\end{aligned}$$\n",
      "\n",
      "Et maintenant, on peut voir la correspondance entre nos deux sommations :\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(B \\cup A_{n+1}) &= p_1 + \\left[\\sum_{k=2}^{n} (-1)^{k+1} p_{k}\\right] + \\mathbb{P}(A_{n+1}) - \\left[\\sum_{k=1}^{n-1} (-1)^{k+1} q_k\\right] - (-1)^{n+1} q_n \\\\\n",
      "  &= p_1 + \\mathbb{P}(A_{n+1}) + \\left[\\sum_{k=2}^{n} (-1)^{k+1} p_{k}\\right] - \\left[\\sum_{k=2}^{n} (-1)^{k} q_{k-1}\\right] - (-1)^{n+1} q_n \\\\\n",
      "  &= p_1 + \\mathbb{P}(A_{n+1}) + \\left[\\sum_{k=2}^{n} (-1)^{k+1} p_{k}\\right] + \\left[\\sum_{k=2}^{n} (-1)^{k+1} q_{k-1}\\right] - (-1)^{n+1} q_n \\\\\n",
      "  &= \\left[\\sum_{i=1}^n \\mathbb{P}(A_i)\\right]- \\left[\\sum_{k=2}^{n} (-1)^{k+1} (p_{k} + q_{k-1})\\right] - (-1)^{n+1} \\mathbb{P}(A_{1} \\cap \\cdots \\cap A_{n+1}).\n",
      "\\end{aligned}$$\n",
      "\n",
      "Soit\n",
      "\n",
      "$$\\begin{aligned}\n",
      "r_k &= p_k + q_{k-1} \\\\\n",
      "  &= \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq n} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_k}) +\n",
      "     \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_{k-1} \\leq n} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_{k-1}} \\cap A_{n+1}) \\\\\n",
      "  &= \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq {n+1}} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_k}).\n",
      "\\end{aligned}$$\n",
      "\n",
      "Cela nous donne\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\mathbb{P}(\\cup_{m=1}^{n+1} A_m) &= r_1 - \\left[\\sum_{k=2}^n (-1)^{k+1} r_k\\right] - (-1)^{n+1} r_{n+1} \\\\\n",
      "  &= (-1)^{1+1} r_1 - \\left[\\sum_{k=2}^n (-1)^{k+1} r_k\\right] + (-1)^{(n+1)+1} r_{n+1} \\\\\n",
      "  &= \\sum_{k=1}^{n+1} (-1)^{k+1} r_k,\n",
      "\\end{aligned}$$\n",
      "\n",
      "et on peut faire l'induction."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Exercice 4 : Distribution al\u00e9atoire du courrier\n",
      "\n",
      "On va commencer avec un peu d'exploration, et trouver les valeurs correctes pour plusieurs $n$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from itertools import permutations\n",
      "\n",
      "# Les lettres et les bo\u00eetes ont les indices 0, 1, 2, etc.\n",
      "# Une lettre est dans bo\u00eete correcte quand les deux ont\n",
      "# la m\u00eame indice.\n",
      "def au_moins_une_lettre_correcte(lst):\n",
      "    for ind, val in enumerate(lst):\n",
      "        if ind == val:\n",
      "            return True\n",
      "    return False\n",
      "\n",
      "# Quand il y a N bo\u00eetes, quelle fraction des distributions sont correctes?\n",
      "def compter_distributions(n):\n",
      "    distributions = list(permutations(range(0,n)))\n",
      "    correctes = [d for d in distributions if au_moins_une_lettre_correcte(d)]\n",
      "    return [len(correctes), len(distributions)]\n",
      "\n",
      "map(compter_distributions, [1,2,3,4,5,6])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\begin{bmatrix}1, & 1\\end{bmatrix}, & \\begin{bmatrix}1, & 2\\end{bmatrix}, & \\begin{bmatrix}4, & 6\\end{bmatrix}, & \\begin{bmatrix}15, & 24\\end{bmatrix}, & \\begin{bmatrix}76, & 120\\end{bmatrix}, & \\begin{bmatrix}455, & 720\\end{bmatrix}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAoAAAAAZBAMAAABTFYKyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 9,
       "text": [
        "[[1, 1], [1, 2], [4, 6], [15, 24], [76, 120], [455, 720]]"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "H\u00e9las, il n'y a pas de motif simple ici."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "list(permutations(range(0,2)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\begin{pmatrix}0, & 1\\end{pmatrix}, & \\begin{pmatrix}1, & 0\\end{pmatrix}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKwAAAAaBAMAAADYjKqDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iMkTNEJl2ie9U\nZqsET4rQAAACcklEQVRIDaWWv2sUQRTHv7ndvdlc7uJpZeWPU1BE4uUfSM7CyiL5C8x1NiKnhYgi\n2UIUU4QrI0JylZguhXaCW1oIuV7QE+xUBH8rIr55s+vuvJ0RwYG7ffN93/nMuzezcNjX6QLhMjxj\n6NbPuGVA+6POCRzShtP6yznOOdWw7ZRJNP4Fgz3vc6ExdqXqA5eqNeM32JhKv7p9s2LVux2rqCTc\noU8FrdY2epnfYBdp8wu43BWE+a8kbAhRTwNKBAdlxfVUHcn8BvsAaO2g0bcJl9Y1dtEWedZIoPa/\nl9jnwP3Mb7CPgakJmppSHtNaqI3LkolnUnpuSewBYHdg/IwNafmeCaKfAsDYuC1Umi5pYgX7C1hJ\nwX7GUgOwmiD6LgCMnaakHPRzq1j1jbBDsJ+xcZ9aMkL4QyxnbCRbQ6Zb2iirDWj50gjsZ+wU/U4v\nNvwkNqPp3r9g2c/Y+oSaQPs4mxB8rmIvurCKqqUmsJ+xtYSOLEHTeWSBVAl53IUF9XY3Bfv/VFtr\noyW7yL3952pxGHg5KFWrm0AHqE/OGowNP1oaT5xNwD0+S/ZztbN0ZDiJ6z31xUJ4b8IrbaObYPtr\nqXqI0k3gq3bt7mvg1KDEjd98eMdvNcR6XRfWn8yNbb/a3uxlfq62uJpRt4Q1oXnLrP2wku/u8zMW\nj3JYKw+KZ32oY3v9zHKW9/kNdi2n3MiD4rnKldnr41GW9/kN9mxOGedB8XzBob0+3MnyPr/BzqYF\nRkZHWRDr30pXMWe/wap+IYuoNRICT6+4RNaM32Dx1Gubd2aaiVMm0fgX0NH/E5qpz3bbnXjmlgHt\njzpzvvR/6r8B46udw7SgL0YAAAAASUVORK5CYII=\n",
       "prompt_number": 10,
       "text": [
        "[(0, 1), (1, 0)]"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "list(permutations(range(0,3)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\begin{pmatrix}0, & 1, & 2\\end{pmatrix}, & \\begin{pmatrix}0, & 2, & 1\\end{pmatrix}, & \\begin{pmatrix}1, & 0, & 2\\end{pmatrix}, & \\begin{pmatrix}1, & 2, & 0\\end{pmatrix}, & \\begin{pmatrix}2, & 0, & 1\\end{pmatrix}, & \\begin{pmatrix}2, & 1, & 0\\end{pmatrix}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAwoAAAAaBAMAAADlKLYUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nkKgAQdhHlUoIRHPxOLL4f+y/rsDfpcVIFnJKsEcAAAAASUVORK5CYII=\n",
       "prompt_number": 11,
       "text": [
        "[(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0)]"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "list(permutations(range(0,4)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\begin{pmatrix}0, & 1, & 2, & 3\\end{pmatrix}, & \\begin{pmatrix}0, & 1, & 3, & 2\\end{pmatrix}, & \\begin{pmatrix}0, & 2, & 1, & 3\\end{pmatrix}, & \\begin{pmatrix}0, & 2, & 3, & 1\\end{pmatrix}, & \\begin{pmatrix}0, & 3, & 1, & 2\\end{pmatrix}, & \\begin{pmatrix}0, & 3, & 2, & 1\\end{pmatrix}, & \\begin{pmatrix}1, & 0, & 2, & 3\\end{pmatrix}, & \\begin{pmatrix}1, & 0, & 3, & 2\\end{pmatrix}, & \\begin{pmatrix}1, & 2, & 0, & 3\\end{pmatrix}, & \\begin{pmatrix}1, & 2, & 3, & 0\\end{pmatrix}, & \\begin{pmatrix}1, & 3, & 0, & 2\\end{pmatrix}, & \\begin{pmatrix}1, & 3, & 2, & 0\\end{pmatrix}, & \\begin{pmatrix}2, & 0, & 1, & 3\\end{pmatrix}, & \\begin{pmatrix}2, & 0, & 3, & 1\\end{pmatrix}, & \\begin{pmatrix}2, & 1, & 0, & 3\\end{pmatrix}, & \\begin{pmatrix}2, & 1, & 3, & 0\\end{pmatrix}, & \\begin{pmatrix}2, & 3, & 0, & 1\\end{pmatrix}, & \\begin{pmatrix}2, & 3, & 1, & 0\\end{pmatrix}, & \\begin{pmatrix}3, & 0, & 1, & 2\\end{pmatrix}, & \\begin{pmatrix}3, & 0, & 2, & 1\\end{pmatrix}, & \\begin{pmatrix}3, & 1, & 0, & 2\\end{pmatrix}, & \\begin{pmatrix}3, & 1, & 2, & 0\\end{pmatrix}, & \\begin{pmatrix}3, & 2, & 0, & 1\\end{pmatrix}, & \\begin{pmatrix}3, & 2, & 1, & 0\\end{pmatrix}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "[(0, 1, 2, 3), (0, 1, 3, 2), (0, 2, 1, 3), (0, 2, 3, 1), (0, 3, 1, 2), (0, 3, \n",
        "2, 1), (1, 0, 2, 3), (1, 0, 3, 2), (1, 2, 0, 3), (1, 2, 3, 0), (1, 3, 0, 2), (\n",
        "1, 3, 2, 0), (2, 0, 1, 3), (2, 0, 3, 1), (2, 1, 0, 3), (2, 1, 3, 0), (2, 3, 0,\n",
        " 1), (2, 3, 1, 0), (3, 0, 1, 2), (3, 0, 2, 1), (3, 1, 0, 2), (3, 1, 2, 0), (3,\n",
        " 2, 0, 1), (3, 2, 1, 0)]"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Essayons les cycles de permutations:\n",
      "\n",
      "<pre>\n",
      "(1)(2)\n",
      "\n",
      "(1)(23)\n",
      "(2)(13)\n",
      "(3)(12)\n",
      "(1)(2)(3)\n",
      "\n",
      "(1)(234)\n",
      "(1)(243)\n",
      "(2)(134)\n",
      "(2)(143)\n",
      "(3)(124)\n",
      "(3)(142)\n",
      "(4)(123)\n",
      "(4)(132)\n",
      "(1)(2)(34)\n",
      "(1)(3)(24)\n",
      "(1)(4)(23)\n",
      "(2)(3)(14)\n",
      "(2)(4)(13)\n",
      "(3)(4)(12)\n",
      "(1)(2)(3)(4)\n",
      "</pre>\n",
      "\n",
      "Il faut compter les permutations avec les points fixes. Mais j'ai perdu plusieurs heures sans trouver la solution avant d'aller en ligne. Apparament, on peux utiliser la formule d\u2019inclusion-exclusion du troisi\u00e8me exercice pour cette t\u00e2che. Je recommence \u00e0 partir de \u00e7a.\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\Omega    &= \\text{ permutations de }n\\text{ objets} \\\\\n",
      "|\\Omega|  &= n! \\\\\n",
      "\\sigma(i) &= \\text{ la lettre qui est d\u00e9pos\u00e9e dans bo\u00eete }i \\\\\n",
      "A_i       &= \\text{ les permutations ou }\\sigma(i) = i \\\\\n",
      "\\mathbb{P}(\\cup_{m=1}^n A_m) &= \\sum_{k=1}^n (-1)^{k+1} p_k \\\\\n",
      "p_k &= \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq n} \\mathbb{P}(A_{i_1} \\cap \\cdots \\cap A_{i_k}) \\\\\n",
      "    &= \\sum_{1 \\leq i_1 \\lt \\cdots \\lt i_k \\leq n} \\frac{(n-k)!}{|\\Omega|} \\\\\n",
      "    &= {n \\choose k} \\frac{(n-k)!}{n!} \\\\\n",
      "    &= \\frac{n!}{(n-k)!k!} \\cdot \\frac{(n-k)!}{n!} \\\\\n",
      "    &= \\frac{1}{k!} \\\\\n",
      "\\mathbb{P}(\\cup_{m=1}^n A_m) &= \\sum_{k=1}^n (-1)^{k+1} \\frac{1}{k!} \\\\\n",
      "\\mathbb{P}(\\cup_{m=1}^4 A_m) &= \\frac{1}{1!} - \\frac{1}{2!} + \\frac{1}{3!} - \\frac{1}{4!} \\\\\n",
      "\\end{aligned}$$\n",
      "\n",
      "### Verfication\n",
      "\n",
      "On veut voir 4, 15, 76, 455, comme on a trouver avec le code source ci-dessus."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(frac(1,1) - frac(1,2) + frac(1,6) - frac(1,24)) * 24"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$15$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAzRAiu5mrdu/dZjJEieXQ\nwQQAAABpSURBVAgdY2BgEGJgYCzijmFgYDB5xMDA8e5dAQODShiQyXitGCjIwA5ksoEYmMxVpRug\noiwKzG+gTCDliWDGGUC0JTIw3DsAYTYyMMQpQJgBDAwToWpNGbiaGRhYM15mMTDHpgLlEQAALDUb\nKItUgjgAAAAASUVORK5CYII=\n",
       "prompt_number": 13,
       "text": [
        "15"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import binomial as bn, factorial as fact\n",
      "\n",
      "def probabilite_au_moins_une_correcte(n):\n",
      "  somme = 0\n",
      "  for k in range(1,n+1):\n",
      "    somme += (-1)**(k+1) * frac(1, fact(k))\n",
      "  return somme\n",
      "\n",
      "[fact(i) * probabilite_au_moins_une_correcte(i) for i in range(1,7)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}1, & 1, & 4, & 15, & 76, & 455\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPYAAAAZBAMAAADwLs6sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90izRCZq3bvZkQy\niVRNeA2AAAACXUlEQVRIDeWVP2zTQBTGvzSN2zhxWlViREotpE4IsxSJBTMgIUSlIDEAEkoXYICq\nHRAtEQMSfzakIGCFjMAUiYUxU+cylSWiGzN/QoEO5vx8yHfv2VKNBBnw8GJ/973v5zufHcz5AcZx\n1PzDaI4DTMxj42efEpOXCjazTJMP6tdtvb6wMhwCw0cDW1dXKuF0a7OfFCTzPjriNqkA81mmahTd\ntXU3iqJdnO87oa2rK5XwIvquS8I+d42zpQJMP2ZZZJrcvsdkR+3dNSyhNMsGKGFpQw1T0fOe5mxI\nBcdPiizVNsVFeIA3cL8IHZSwFutUirC39slW2W9R6maw44Q/YrtBNvv1w4HAzGLm7Hu2BQFK6Ly8\nAVApMO8LyGSXWuU9zvZ6aJ9BdYfplHAAJ/qgUoC9lc1W+XOMgTbQHsF5xnSdUGoqPS7JOyZ3llAa\ng1z2aqDCzOM+MNOEy9bjd4ITb8O47Jt9Gdnsm8CHvglW5/NANYT705YpYSqEt0elAPvj+vqPDTuM\nXsSnwGrL1iu7ak27Yt6UoO7JGVEpwFbx/NtC7GXgkI1G/RvQUM+7y/Q4wQEmQiomux2a1vh52wrw\nCZWvpofYi3CfMGdDsXEQF3eYPU4oL+NVQEWzndufO5gwv21SAd5FHRwxl5dM5ZVbLbsX5eeK7V29\nA9ueJCxuXwGo6L2mvMAbqmaRSi0wx9Nz6aSxPHvSaPx/99IofSYVV3gSQTpJz7MnTSm7IvqlgktJ\nF68ZTrLk2HV7ypb3KBX0dBv7yXCSI8euu1M2i/sHl/8v2/dz3pu/vOo1f+EXGkawrgVEX8UAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 14,
       "text": [
        "[1, 1, 4, 15, 76, 455]"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Le voil\u00e0 !\n",
      "\n",
      "### R\u00e9ponses\n",
      "\n",
      "La probabilit\u00e9 d'une distribution correcte est $\\frac{1}{n!}$. La probabilit\u00e9 qu'au moins une lettre parvienne au bon destinataire est donn\u00e9e par $\\mathbb{P}(\\cup_{m=1}^n A_m)$ ci-dessus. La probabilit\u00e9 qu'aucune lettre n'arrive au bon destinataire est $1-\\mathbb{P}(\\cup_{m=1}^n A_m)$. Et $d_n$ est \u00e9gal \u00e0 $n!(1-\\mathbb{P}(\\cup_{m=1}^n A_m))$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    }
   ],
   "metadata": {}
  }
 ]
}