{
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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "source": [
    "1回の試行で14試合5口分の予想が前回購入のものと一致する確率は\n",
    "\n",
    "$ p = \\left( \\frac{1}{3} \\right) ^{70} $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "$N$ 回試行したときに1度も上記事象が発生しない確率\n",
    "\n",
    "$ q = \\left( 1 - p \\right) ^N $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "$ q $ の対数を取る\n",
    "\n",
    "$ \\log{q} = N \\log \\left( 1 - p \\right) $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "1次近似すると\n",
    "\n",
    "$ \\log{q} \\sim -N \\left( \\frac{1}{3} \\right) ^{70} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def prob(k):\n",
    "    lq = - 1.0 / np.power(3.0, 70 - k)\n",
    "    q  = np.exp(lq)\n",
    "    return q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def probs(ks):\n",
    "    lab = []\n",
    "    val = []\n",
    "    for k in ks:\n",
    "        lab.append(k)\n",
    "        val.append(prob(k))\n",
    "    df = pd.DataFrame({'q': val})\n",
    "    df.index = lab\n",
    "    return df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "$ N = 3 ^k $ 回試行したときの確率の概算値"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>q</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>55</th>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>60</th>\n",
       "      <td>0.999983</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>65</th>\n",
       "      <td>0.995893</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>70</th>\n",
       "      <td>0.367879</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "           q\n",
       "50  1.000000\n",
       "55  1.000000\n",
       "60  0.999983\n",
       "65  0.995893\n",
       "70  0.367879"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "probs([50, 55, 60, 65, 70])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "1回の抽選で400万口程度売れるらしいので、1回の抽選で $ 3 ^{12} $ 以上の試行がある"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "800000.0\n",
      "531441.0\n"
     ]
    }
   ],
   "source": [
    "print(4000000 / 5)\n",
    "print(np.power(3.0, 12))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "0.5%で当該事象の起こるくらいに必要な抽選回数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.4610818892266729e+24"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.power(3.0, 65 - 12 - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "source": [
    "誕生日のパラドックスのような状況になるには、試行1回当りの発生確率が小さすぎるようだ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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