{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Marche aléatoire"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from random import random\n",
    "\n",
    "def freqDroite1(n):\n",
    "    \"\"\"Fréquence de marches aléatoires avec exactement \n",
    "    1 déplacement vers la droite sur un échantillon de taille n\"\"\"\n",
    "    c = 0\n",
    "    for i in range(1, n + 1):\n",
    "        d = 0\n",
    "        for j in range(1, 5):\n",
    "            if random() <= 0.6:\n",
    "                d = d + 1\n",
    "        if d == 1:\n",
    "            c = c + 1\n",
    "    return c / n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.15360000000000004\n"
     ]
    }
   ],
   "source": [
    "## Probabilité d'avoir exactement un déplacement vers la droite\n",
    "p = 4 * 0.6 * 0.4 ** 3\n",
    "print(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "taille_echantillon = np.arange(0, 20001, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([     0,    500,   1000,   1500,   2000,   2500,   3000,   3500,\n",
       "         4000,   4500,   5000,   5500,   6000,   6500,   7000,   7500,\n",
       "         8000,   8500,   9000,   9500,  10000,  10500,  11000,  11500,\n",
       "        12000,  12500,  13000,  13500,  14000,  14500,  15000,  15500,\n",
       "        16000,  16500,  17000,  17500,  18000,  18500,  19000,  19500,\n",
       "        20000,  20500,  21000,  21500,  22000,  22500,  23000,  23500,\n",
       "        24000,  24500,  25000,  25500,  26000,  26500,  27000,  27500,\n",
       "        28000,  28500,  29000,  29500,  30000,  30500,  31000,  31500,\n",
       "        32000,  32500,  33000,  33500,  34000,  34500,  35000,  35500,\n",
       "        36000,  36500,  37000,  37500,  38000,  38500,  39000,  39500,\n",
       "        40000,  40500,  41000,  41500,  42000,  42500,  43000,  43500,\n",
       "        44000,  44500,  45000,  45500,  46000,  46500,  47000,  47500,\n",
       "        48000,  48500,  49000,  49500,  50000,  50500,  51000,  51500,\n",
       "        52000,  52500,  53000,  53500,  54000,  54500,  55000,  55500,\n",
       "        56000,  56500,  57000,  57500,  58000,  58500,  59000,  59500,\n",
       "        60000,  60500,  61000,  61500,  62000,  62500,  63000,  63500,\n",
       "        64000,  64500,  65000,  65500,  66000,  66500,  67000,  67500,\n",
       "        68000,  68500,  69000,  69500,  70000,  70500,  71000,  71500,\n",
       "        72000,  72500,  73000,  73500,  74000,  74500,  75000,  75500,\n",
       "        76000,  76500,  77000,  77500,  78000,  78500,  79000,  79500,\n",
       "        80000,  80500,  81000,  81500,  82000,  82500,  83000,  83500,\n",
       "        84000,  84500,  85000,  85500,  86000,  86500,  87000,  87500,\n",
       "        88000,  88500,  89000,  89500,  90000,  90500,  91000,  91500,\n",
       "        92000,  92500,  93000,  93500,  94000,  94500,  95000,  95500,\n",
       "        96000,  96500,  97000,  97500,  98000,  98500,  99000,  99500,\n",
       "       100000])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taille_echantillon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "freqDroite1Vect = np.vectorize(freqDroite1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(taille_echantillon[1:], freqDroite1Vect(taille_echantillon[1:]), s=4)\n",
    "plt.xlabel(\"Taille de l'échantillon\")\n",
    "plt.ylabel(\"Fréquence\")\n",
    "plt.savefig(\"marche-aleatoire.pdf\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probleme des partis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def proba_parti(a, b):\n",
    "    \"\"\"Probabilité que le joueur 1 gagne s'il lui reste\n",
    "    a points à marquer et b points pour le joueur b\"\"\"\n",
    "    if a == 0:\n",
    "        return 1\n",
    "    if b == 0:\n",
    "        return 0\n",
    "    if a == b:\n",
    "        return 0.5\n",
    "    return 0.5 * proba_parti(a - 1, b) + 0.5 * proba_parti(a, b - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.875"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "proba_parti(1, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.875"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "7/8"
   ]
  }
 ],
 "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": "3.6.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}