Does picking the same numbers every time increase one's chances of winning the lottery.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# [Does picking the same numbers every time increase one's chances of winning the lottery?](https://www.reddit.com/r/probabilitytheory/comments/g772eo/does_picking_the_same_numbers_every_time_increase/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "\n",
    "from joblib import parallel,delayed\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "sns.set_style('whitegrid')\n",
    "sns.set_context('poster')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# let's just look at a 3-choice game"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.33527\n"
     ]
    }
   ],
   "source": [
    "n = 3\n",
    "pick = 1\n",
    "n_sim = int(1e5)\n",
    "games = np.random.choice(np.arange(n),size = n_sim,)\n",
    "p_win = np.sum(games == pick) / n_sim\n",
    "print(p_win)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.33435\n"
     ]
    }
   ],
   "source": [
    "n = 3\n",
    "n_sim = int(1e5)\n",
    "games = np.random.choice(np.arange(n),size = n_sim,)\n",
    "picks = np.random.choice(np.arange(n),size = n_sim,)\n",
    "p_win = np.sum(games == picks) / n_sim\n",
    "print(p_win)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fixed(n,n_sim = int(1e5)):\n",
    "    pick = 1\n",
    "    games = np.random.choice(np.arange(n),size = n_sim,)\n",
    "    p_fix = np.sum(games == pick) / n_sim\n",
    "    return p_fix\n",
    "\n",
    "def random(n, n_sim = int(1e5)):\n",
    "    games = np.random.choice(np.arange(n),size = n_sim,)\n",
    "    picks = np.random.choice(np.arange(n),size = n_sim,)\n",
    "    p_random = np.sum(games == picks) / n_sim\n",
    "    return p_random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "results = dict(n = [],\n",
    "               p = [],\n",
    "               fixed = [],\n",
    "               )\n",
    "for n in np.arange(3,int(1e4) + 1,):\n",
    "    # fixed choice\n",
    "    #for _ in range(int(1e2)):\n",
    "    p_fix = fixed(n)\n",
    "    results['n'].append(n)\n",
    "    results['p'].append(p_fix)\n",
    "    results['fixed'].append(True)\n",
    "    \n",
    "    # random pick\n",
    "    #for _ in range(int(1e2)):\n",
    "    p_random = random(n)\n",
    "    results['n'].append(n+.1)\n",
    "    results['p'].append(p_random)\n",
    "    results['fixed'].append(False)\n",
    "results = pd.DataFrame(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
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/9NBDnHfeeaxdu5Y99tiDd77znTzzzDP86Ec/4r777uNHP/oRO+6447Z46f0YqCuY30MtSZIkqdj+9Kc/cfPNN7PTTjtxyy23MGnSpN5j3//+97n88su56aabBg3UP/nJT3jllVc466yz+PSnP927/8EHH+Scc87h29/+dm+gXrFiBZ/61Kfo6uri2muv5aijjgIgTVOuv/56vvWtb3HhhRdyyy23DNMr3jgDdQVLN3JbkiRJ0jrfmfcC37rvGVo7s6UuZbM11lbxz0fuzt8fukvR2nzttdcIIWz0+A9/+EPmzJlDS0sLRx99NMcff3y/MA3w0Y9+lMsvv5zFixcP+lzLli0DYLvttuu3/7DDDuPSSy9lwoQJ5HI5MpkMP/vZz1i9ejVnnnlmb5gGSJKEc889l/vvv5/HHnuMxx57jP32229LX/ZWMVBLkiRJGtW+89sXRlSYBmjtzPKd375Q1EC9qa/NmjJlCgAHH3wwBx98cL9jHR0dvPDCCzz22GMkSUJXV9egz3XAAQdw88038/Wvf51nn32W9773vRx00EE0NjbysY99rN+5jz76KABz5swZsK1DDjmEP//5zyxYsMBArSJylW9JkiRpk/7+PbuMyB7qv39P8cI0bNnXZrW3t3PbbbfxwAMP8Nxzz/HGG2+QpilJkpBuxnTT97///SxcuJCf/OQn3Hrrrdx6663U1NRwwAEHcMIJJ/CBD3yA6up8XH399dcBOPfccwdts+e8bclAPVo4h1qSJEka0N8fuktRe3or3RtvvMFpp53GK6+8wrhx45g1axbHHnsse+yxBwceeCBHHHEE2eymP5y44IIL+NjHPsa8efN46KGH+OMf/8gjjzzCI488wm233cb3v/99amtre9s64ogjaGho2Gh7e+yxR9Fe4+YyUEuSJEmSNtu3vvUtXnnlFU455RQuvfRSampqeo+1tLRsVpjuscMOO3D22Wdz9tln09HRwbx587jkkktYuHAh9957L+9///uZNm0aL730Eh//+MfZf//9h+MlDVmm1AVo+PQf8G0PtSRJkqSt98QTTwBw9tln9wvTAA8//HDv7Vwut9E2LrzwQt73vvexZMmS3n11dXUcddRRfOADHwDoXdisJ0TPmzdvwLYuuugiTjnlFO6///4hvJqtY6CuZH3mUDviW5IkSVIxzJgxA4AHHnig3/7HH3+cyy67rPd+R0fHRtuYMmUKq1at4uqrr6azs7N3f3Nzc29w3nvvvQH4m7/5G8aMGcONN97Ivffe26+d22+/nTvuuINnnnmGffbZZ+te2BA45FuSJEmStNlOP/10HnroIb72ta9xzz33sP3227No0SL+8pe/MH78eKZOncqyZctYvnw5O+6444BtfOITn+DXv/41v/rVr3jsscfYa6+9yGazPP7446xevZrjjz+eAw88EIDp06dz+eWXc+GFF3Leeeex++67s/POO/Pyyy8TYySTyXDllVf2rkK+LdlDXcFSXOVbkiRJUnEddthh3HDDDcyePZtXXnmFhx9+mK6uLk499VT++7//m2OPPRbYsAe7rwkTJvDd736Xj3zkI9TW1vLb3/6WBQsWsPPOO3PppZfy9a9/vd/5xx13HLfeeisnnHACK1eu5H//939pbm7mmGOO4dZbb+W4444b1te8MfZQVzDnUEuSJEnalDlz5hBj3KLHHHbYYRx22GEDHrv44ou5+OKLe+/PnDlzwPanTp3KRRddRGNj42Y959vf/vYNgnap2UM9SmzOd8FJkiRJkjafgVqSJEmSpCEwUFeyxDnUkiRJkjRcDNQVrG+cThzyLUmSJElFZaCuYH1X+TZOS5IkSVJxGaglSZIkSRoCA3Ulcw61JEmSJA0bA/Vo4RxqSZIkSSoqA7UkSZIkSUNgoK5ofYd820MtSZIkScVkoK5k/aZQG6glSZIkqZgM1JIkSZIkDYGBuqL1+R5qO6glSZIkqagM1KOGiVqSJEmSiqm61AVo27CHWpIkSdL6rrnmGq699totekyMcZiqGXkM1BUs2chtSZIkSQIIIXDiiSf227do0SIee+wxJk+ezMEHH1yiykYGA3UlS4zRkiRJkjbu6KOP5uijj+637/bbb+exxx5j11135aqrripRZSODc6grmt9DLUmSJEnDxUA9ShinJUmSJBXDokWLCCHwyU9+krvuuotDDz2UWbNmccopp9DV1cXcuXMJIfCHP/xhg8ded911hBC45pprNjg2b948Pv7xj3PAAQcwa9YsTjzxRL773e/S2dm5LV7WkDjkW5IkSZK0xZ566inuu+8+Zs2axW677UZjYyM1NTVDauu6667j6quvpqamhlmzZjFp0iQWLlzIlVdeyYMPPsh//dd/UVtbW+RXsPUM1BUs7Tvk22W+JUmSJBXRq6++yplnnslFF10EQC6XG1I7jzzyCFdffTXbb789N9xwA7vtthsAbW1tXHDBBTzwwANce+21fOpTnypa7cVioK5g/dckM1BLkiRJA3rkGvjNFdDZUupKNl9tE7z3s3Dw+SUt4/TTT++9nckMbUbxd7/7XQAuvvji3jAN0NDQwJe//GUOP/xwfvzjH3PeeeeVXS+1c6glSZIkjW6PXDuywjTk631ky74/utjGjh3LDjvssFVtZLPZ3rnWc+bM2eD4pEmT2HPPPWlpaeGpp57aqucaDvZQV7D+HdT2UEuSJEkDOvi8kdlDffB5JS1h/PjxW93G6tWraW9vB2D27NmDnvv666+z7777bvVzFpOBuoKlJL2h2jgtSZIkbcTB55d86PRINJQh3uvPs85mswCMGTOGI488ctDHTp06dYufb7gZqCVJkiRJRZMUFnPqCct9rVmzpt/9CRMmUFNTQ3d3N1/96lepqqraJjUWi3OoK1n/VckkSZIkadg1NDQAsHz58g2OPfHEE/3u19TUsM8++9DV1cX8+fM3OL+zs5OTTz6Zv/3bv2XRokXDU/BWMFBXMBf5liRJkrSthRAAuPnmm+nq6urd/+Mf/5jHH398g/PPOOMMAC655BKeeeaZ3v3d3d1cdtll/OUvf6GtrY2ZM2cOc+VbziHfkiRJkqSi+chHPsJNN93EggULOOaYY9hrr7144YUXeO655zjxxBO56667+p1/9NFHc8YZZ/CDH/yAk08+mb322ospU6bw5JNP8vrrrzNp0iS+8Y1vlOjVDM4e6lHDLmpJkiRJw2/mzJnccsstHHXUUTQ3NzNv3jzGjRvHd77zHU466aQBH/O5z32Ob3/72xxwwAE8//zzzJs3j/r6eubOncudd97JLrvsso1fxeaxh7qi9R30baCWJEmStGknn3wyJ5988kaPz5w5kxjjoG3stttuXHvtwN+T3fPY1tbWfvuPPPLITa70XW7soZYkSZIkaQgM1JWs7yrfdlBLkiRJUlEZqEcNE7UkSZIkFZOBepQwTkuSJElScRmoR4nESC1JkiRJRWWgrmR951BLkiRJkoqqbL82K4RwJPA5YBZQCywErogx/moL2jgO+GfgAKAJeB34JfDvMcZFRS+67PRdlMweakmSJEmjW1rkXFSWPdQhhDOBe4GDgd8D84FDgHtCCOdsZhufBe4GjgRi4TbAJ4A/hhD2KHLZ5cdFviVJkjTKJYVRm9lstsSVqBz0BOqkSKN5yy5QhxBmANcDq4H9Y4zHxxiPIR+o1wBXhxB22EQbewL/DrQA744xHhRj/CDwNuA6YCrwvWF8GZIkSZLKQH19PQBr1qwpcSUqB62trQDU1dUVpb2yC9TA+UAd8M0Y45M9O2OMC4ArgXpgU73Uc4Eq4Bsxxvl92ugiPwR8GXBgCGGnItdeZpxDLUmSpNFt4sSJALzxxhssW7aM9vZ2crlc0Yf+qnylaUoul6O5uZklS5YAMHbs2KK0XY5zqI8tbO8c4Ngd5HuejwMuGaSNTuBPwLz1D8QYu0IIL5Lvpd4eeHmrqh0p/IUhSZKkUWjcuHG0t7ezYsUKli9fzvLly0tdkgpyuRwAmcy27edtaGhg0qRJRWmrrAJ1CCEB9gRywNMDnPJM4dg7QghJjHHAlBhjvISNBO4QQmPhOQBGwcJkkiRJ0uiVJAnbbbcdjY2NrFmzhra2Nrq7u+2hLgPt7e1APuAOtyRJqKurY+zYsUyaNKloIb6sAjUwkfxw72Uxxs71D8YYu0MIy4FpwFjyc6q31GfIr/i9IMb46tYU26OlpYWFCxcWo6miWr58OVMLt5e+8UZZ1qjRxWtQ5cDrUOXCa1HlwOtQ5aCtrW2bPE9raysrVqzg5ZeLN0i53OZQNxa2g/1E1xa2TVvaeAjhePJfxZUDPr2ljx9x/B5qSZIkSRo25dZDnStsBxt/kay33SwhhPcDPyO/WNlnY4y/2eLqNqKpqYkQQrGaK5o/LPod5OfcM3XaNGbPnl3agjRq9Xz67TWoUvI6VLnwWlQ58DpUORgJ12GMkZaWlo0eL7ce6p5KxwxyTn1h27q5jYYQPk5+kbN64Esxxq8OrbyRJXWVb0mSJEkaNuUWqNeQD9VTQggb9J4X9k0B2mOMqzanwRDCZcB3yfdM/0thwbJRof+IbxddkCRJkqRiKqtAXVi1+yny4Xf3AU4J5Gv+86baCiEkIYT/Ai4GOoCPxRi/VcRyJUmSJEmjWFkF6oJ7CtsPDnCsZ9/dm9HO14GzyPd6HxNj/H9FqG1ESfoN+baHWpIkSZKKqRwD9feAduAzIYTe2ekhhP3Jr8y9Friuz/5dQwh7hBDG99l3LPAvQDfw/hjjg9uq+HKSbvSOJEmSJGlrldsq38QYXwohXAB8G5gfQrif/Ire7yNf7+kxxqV9HnI/sBPwd8D3C/u+WNi+AfxDCOEfNvJ0X44xPl3cVyBJkiRJGg3KLlADxBivCyG8Qr5H+j3k50A/RD4A3z/YY0MIDcABhbs7AP9nkNP/C6jgQO0q35IkSZI0XMoyUAPEGH8B/GIzztt5vftt5Bc1G/WSpO9Ib8d8S5IkSVIxleMcag2D1DwtSZIkSUVloB4lEnuoJUmSJKmoDNQVLHUOtSRJkiQNGwN1BUuSPoHaMd+SJEmSVFQG6oq2LlAbpyVJkiSpuAzUkiRJkiQNgYG6kjmFWpIkSZKGjYF6tHAOtSRJkiQVlYG6gtlBLUmSJEnDx0A9WthDLUmSJElFVV2MRkII+wJvA8YDy4GnYozPFqNtDV2a2EctSZIkScNlyIE6hJABzgU+C2w/wPGngX+LMd4x9PJULPZPS5IkSVJxDWnIdwghAW4G/gPYAVgL/Al4GPgz0AHsCfwshHBFcUrVlkr6vb1GakmSJEkqpqHOoT4L+AjQDHwcmBhj3C/GeGiMcV9gInAO0AJcGEI4rijVasgSA7UkSZIkFdVQh3x/gnyX5wdjjL9Z/2CMsQP4rxDCIuBu4J+BXw61SA2Rc6glSZIkadgMtYc6AHGgMN1XjPEe4Flg/yE+j4okdZVvSZIkSSqqoQbqdiC3mee24Vcil0afHuok3dy3S5IkSZK0OYYaqO8F3h5COGSwk0IIuwF7A/cP8Xm0NRK/ZlySJEmShstQE9e/Aq8Bd4YQ3j/QCSGEvYCfA6uAzwzxebRV+gwMsIdakiRJkopqqIuS/RvwR+Ak4OchhJeBPwArgQbyX5m1b+Hc14FbQgjrt5HGGOcM8fm1GZI+Q76dQS1JkiRJxTXUQP0PrMtoCbBz4b+BbF/4b31mvG0ozdlDLUmSJEnFNNRA/XdFrULDIslUlboESZIkSapYQwrUMcYfFLsQFV+/pdWdQy1JkiRJReUy0BUsyfR9ex1hL0mSJEnFZKCuYGm/Vb4N1JIkSZJUTAbqCtZ3lW97qCVJkiSpuAzUlazv12blDNSSJEmSVEwG6gpmD7UkSZIkDR8DdSXr20PtHGpJkiRJKioDdQVLknVvb2IPtSRJkiQVlYG6giXYQy1JkiRJw8VAXcn6TaE2UEuSJElSMRmoK9q6t9ceakmSJEkqLgN1Bctk1nVRO4dakiRJkorLQF3J+ixKRporXR2SJEmSVIEM1BWs79dQ2z8tSZIkScVloK5ofRO1kVqSJEmSislAXcGSpN8y3yWrQ5IkSZIqkYG6kvWbQ22gliRJkqRiMlBXMHuoJUmSJGjiwNYAACAASURBVGn4GKgrWeIcakmSJEkaLgbqCpb0W5TMr82SJEmSpGIyUFewJFPVe9v+aUmSJEkqLgN1Bes74juxh1qSJEmSispAXcn6rPJtD7UkSZIkFZeBepSwh1qSJEmSistAXcEyGXuoJUmSJGm4GKgrWb+vobaHWpIkSZKKyUBdwZJk3Srffg+1JEmSJBWXgbqCJUm/LuqS1SFJkiRJlchAXcH6BWp7qCVJkiSpqAzUFc0eakmSJEkaLgbqCpZk7KGWJEmSpOFioK5oyaZPkSRJkiQNiYG6gjmHWpIkSZKGj4G6gmUyfd5eA7UkSZIkFVV1qQvYmBDCkcDngFlALbAQuCLG+KshtheAJ4B/jzH+e9EKLWvrAnVCroR1SJIkSVLlKcse6hDCmcC9wMHA74H5wCHAPSGEc4bQ3lTgDqCuiGWWvaSqat2d1EAtSZIkScVUdoE6hDADuB5YDewfYzw+xngM+UC9Brg6hLDDFrS3F/AQ8PbhqLecpcm6QG0PtSRJkiQVV9kFauB88j3J34wxPtmzM8a4ALgSqAc22UsdQmgIIXwBeBTYHXhxeMotX33nUCdptoSVSJIkSVLlKcdAfWxhe+cAx+4obI/bjHY+ClxKvlf7ZOCHW1/aCJPp00PtkG9JkiRJKqqyCtQhhATYE8gBTw9wyjOFY+8onDuYN4FLgN1ijHds4tyKlPTroTZQS5IkSVIxldsq3xPJD/deFmPsXP9gjLE7hLAcmAaMJd/7PKAY413AXcNVaF8tLS0sXLhwWzzVFmletYymwu2O9rayrFGji9egyoHXocqF16LKgdehysFIvg7LqocaaCxs2wY5Z21h2zTIOQKSZF0nfsYeakmSJEkqqnLroe5Jfekg5yTrbUuuqamJ/Ndcl5c3l73Oi/Pzt+vqapg9e3ZpC9Ko1fOpo9egSsnrUOXCa1HlwOtQ5WAkXIcxRlpaWjZ6vNx6qHsqHTPIOfWFbesw1zLiZTLrPi9xDrUkSZIkFVe5Beo15EP1lBDCBr3nhX1TgPYY46ptXdxIk+mzyjcGakmSJEkqqrIK1DHGFHgKqCL/3dHrC+Rr/vO2rGukylS5yrckSZIkDZeyCtQF9xS2HxzgWM++u7dRLSNaVXWfTn4DtSRJkiQVVTkG6u8B7cBnQgi9s9NDCPsDnya/yvd1ffbvGkLYI4QwfptXWub6DvlO0mwJK5EkSZKkylN2gTrG+BJwATAOmB9C+GUI4R7gEfLfPX1OjHFpn4fcDzwNfGhb11ruqqr6Bmp7qCVJkiSpmMouUAPEGK8DTgR+B7wHOAB4CDgqxnhTKWsbSarWX+U7HezbyCRJkiRJW6Lcvoe6V4zxF8AvNuO8nTezvS8CX9yqokaYJJOQkpCQkgJpmiNJqjb5OEmSJEnSppVlD7WKI0kS0j5vcba7u4TVSJIkSVJlMVBXuDRJem9ncwZqSZIkSSoWA3WF69tDncu6MJkkSZIkFYuBusLl+syZzmbtoZYkSZKkYjFQV7x1Q75zOXuoJUmSJKlYDNQVLk3WvcWpPdSSJEmSVDQG6grXf8h3toSVSJIkSVJlMVBXPFf5liRJkqThYKCucP2HfDuHWpIkSZKKxUBd4dJ+PdQO+ZYkSZKkYjFQV7o+c6hz3Q75liRJkqRiMVBXuP6LknWVsBJJkiRJqiwG6gqXZqp7b2e7DdSSJEmSVCwG6gqX9nmLDdSSJEmSVDwG6grXt4c619VZwkokSZIkqbIYqCtdxjnUkiRJkjQcDNQVLu23yreBWpIkSZKKxUBd6ZI+Q76zfm2WJEmSJBWLgbrCpX2HfHc7h1qSJEmSisVAXen6LEqWOuRbkiRJkorGQF3p+vRQ57L2UEuSJElSsRioK51zqCVJkiRpWBioK12fHurUr82SJEmSpKIxUFe6PnOo/dosSZIkSSoeA3WFy1XV9d5OuttLWIkkSZIkVRYDdYVL+wRqutpKV4gkSZIkVRgDdYVLaup7b6dd9lBLkiRJUrEYqCtcVfW6Huq0a20JK5EkSZKkymKgrnBJTZ851A75liRJkqSiMVBXuKradUO+cVEySZIkSSoaA3WFq+7TQ53pdsi3JEmSJBWLgbrCVfdZlCyTtYdakiRJkorFQF3hqmvH9N7OZDtIc7kSViNJkiRJlcNAXeEy1TWkSVX+dpplbYe91JIkSZJUDAbqUSCbWTePurW1pYSVSJIkSVLlMFCPArlMbe/ttQZqSZIkSSoKA/UokKtetzDZ2jYDtSRJkiQVg4F6FJiYXdF7u6t5eQkrkSRJkqTKYaAeBWqT7t7bY155sISVSJIkSVLlMFCPAtnqxt7bzemYQc6UJEmSJG0uA/Uo8PK0I3tvr+3sKmElkiRJklQ5DNSjQNWY8b23q1sWl7ASSZIkSaocBupRoKZxQu/thuYXIU1LWI0kSZIkVQYD9ShQ17iuh7orm9K9dk0Jq5EkSZKkymCgHgWq6xr73V+8dFmJKpEkSZKkymGgHgWSJOl3/8Xf312iSiRJkiSpchioR4m+mXrt0udLV4gkSZIkVQgD9Sgx7m0H997epevZElYiSZIkSZXBQD1KTNl73XdRt3VmSbs7S1iNJEmSJI18BupRomnXg/rdf+Cum0pUiSRJkiRVBgP1KJFk+r/Vk5/5aYkqkSRJkqTKYKAeRZbWbN/v/qrXXyhRJZIkSZI08hmoR5E5f3dVv/sv/fDcElUiSZIkSSOfgXoUGT9x8gb7HvvNHSWoRJIkSZJGPgP1KLPz3G/3u588ej3tz/xviaqRJEmSpJHLQD3KTNj+bUxsqO237693XMHK2/8VcrkSVSVJkiRJI4+BehTa6fy7Ntj38rN/5vGvHUf64rwSVCRJkiRJI091qQvYmBDCkcDngFlALbAQuCLG+KstaGN34FLg3cBk4DngBuC6GOOo7o7d7ZM/59n/OGmD/U/8vy8DX2bqe/+B7d/1QZIk2fbFSZIkSdIIUJY91CGEM4F7gYOB3wPzgUOAe0II52xmG/sAC4CPAS8D9wA7AtcAPyx+1SNL45g6Zv3jjzZ6fNlvrueJK4/l8a8ewy++/a+sWfoqpOk2rFCSJEmSylvZ9VCHEGYA1wOrgXfHGJ8s7D8AuA+4OoTwPzHG1wZpIyEfmscBc2OMNxX2Ty208X9CCHfEGG8b3ldT3jJjp7Hvx6/h8RvPH/S8mS1/5oXvnb3B/rqpb2WXWYeQ7nEi3VVjaKirJZOxR1uSJEnS6FB2gRo4H6gDLu8J0wAxxgUhhCuBfwfOAS4ZpI2jyA8V/01PmC60sSyE8I/AQ8AngVEdqAGYujv7fuZXdK9t5sn/+PAWPbRj2Ys8ff+LcP9Nmz65oLq2jr9mdqOjaSb7Nq2CtxzE2PGTmPTWWTTV10GS5HvCO1uhrmlLX40kSZIkbTPlGKiPLWzvHODYHeQD9XEMHqg32kaM8eEQwlLg3SGEsTHG5q0ptlJUjxnLvp/JT09/cuFDdN932bA8T3dnB2/jSWh/kq7lwEu/YwWwokjtr800MibXWqTW8lZVTaF16ixCx5OkrW9SU50hl0tpa9yRqd2vs3riXtyf3ZfpNe20rVxM0r6aCVUd7DQ+Q6azhdpd383aMTNY1dzKDrzBgrbtWZtWsf/4ZrYfV0PzmB1Y27KK1sw4aseMpbGxkQm00J2D7voJ1LYuIVNdSzaFXLabidN24KXVWSY21lGVZsmQ0lRfQ3tXlu6qBsZWZ2lpbaO6CqobJpBLqqhLO4CUzqSO2rSDrq5uauob6O7uoj1XRX1NNdlcSntXlvFjakhy3VBdm/9wI82RJhly2W4ymSqyKWRzOWqqqsgkkEL+vCQhSRLSNO3ddnXnqK2pWvfDTFPSNCVHQiZh8Dn6hTa32FAfJ0mSJG2hsgrUhaHaewI54OkBTnmmcOwdIYQkxrixSb3vKGyf3MjxCEwrPNejQ6+4Mu01+90wOx+uc+0ttLy0gL/++rvUrl1W4so2rdhhGmBCdjkTljxAS8+Ozmx+2/4irwOs/gMH8YcNHtdVeEDHshcAaCI/j2H3nuPkJ/f3yADdhXNWD1LP4sK2+K+0v57Am81t/dz5tLuLtYzh8fu6N+v8mqqENIUW6qnLdZCwqTUEEwrRvp/qTEJVJiElobVqPCmQkA/1+S10Z7M05PJvVntmDLVpJy2ZcUxKV5HNpb2tJkB94cOBrkw9Nbl2SCCbS1mbzVCVrnttmapq2jONrM1VMT478MdFY2qqaO/KDlB1oe6qhOZcPQ3pWlYkE+hM6klJ2DFZQnc2ZWUyHsh/iDStKz8DpjOpB2BibVf+9VSPp6tqDA1rX+e1dCqTu98AoK4mw+J0Kp1JHRlSJncvoTrNP6Y7qem9nW+/iYnV7SzJTiBNEtoyjYzLriJJMjR1r6Aqk1Bfk6G1ZhIdVU2FnxSkSabwriSQJKTA1Lb8n4W11eNpqZ2SfzeShGltz/Z7/Wtqp9NePZaEDGnPhyNJ/n3seTd6fm4ZckxtfY7muu3IJtVkM7VUpV1ManuRBFhTN4OO6iZSEtY0N9PckeX1efWMH1PN6rXr3rO66gwNdVVkkgyr2rqorU5YW/iz3lBX3fveA7R35aipSqiuyn+41jOwBmBFayd1NVU01VWRkPT+PAbS+9oGuArSjTxuZWsn2VzKpKZaqjMZOrqzrFnbTVoooL62mjE1VWTTlPrqqv4PTnr/R/9bhfvr7UhIqUq7SZMqUhLSJANJplB7hgltL1HftZq2uql0VjfRkq1heUsn08Y30tLRSRNrGZtrZk399gBUp510J7XkMtUkaY4kzeXbThJqsmvpztSRJn1rTjfy01mvznTdWVW5Lhq73qQ6105HVRNVaRedVY1U5TrpztSRSbsZ1/E6q+p37L0utthm/Erc1CnNzfnP8uc9fuuWP3+f51je3AHA1HF11FQVZ1mc4n0cWT4fbA7vZ6zl8zq3REJK49KFVKfdPPrUIfk/31vXYEVKKuyFJUn+33dFbHGrW1i5aiVpdSMrdpjCpOk7FaGmbS9Jy2ihqRDCJOBNYFmMcdpGznmDfBgeH2Ncs5Fz/gTsDbwjxvjUAMd/CnwU+ECM8edDrXfhwoW/AQ4b6uNHsiTXRUfHWtqXvcC01X/i8bXT2GnFw9TTXurSJEmSJI0gqxt2ZsKh/1TqMjblwdmzZ793/Z1l1UMNNBa2bYOcs7awbQIGDNSb0U7fNjQEaaaG2jE11L5lX9rZlz2AdSPtN607m6Ozu5v2LGS61jJtxQJey02iZc0KGqtTckufJpN2M72mjezawfpr+3szmczk9M0tfj2SJEmSSqOjelypSxiycgvUPeM6B+s2T9bbDqWdzWljszU1NRFCKEZTRbdw4UIAZs+eXeJKNuV97F3qEobbZsztTXPZ3iGVSZLQ2Z0fWtrzqGyav2iTBJI0pb07S1dXF011VVBVS5pL6e5qp6a2HnJddLU1U1NXT5pU05ZWM6YqPyc6k0Cuu4vlHVWMr4PuXEom10WaVJFJEqrIkctUk0uq6G5vobUjS3VNLW82tzNzYh3ZttVk1iwiN/Gt5DK1tHd0kNQ2MC5tpiOtoSpJaG5toSHpZG1Sz5PxeRqqYc999idd8SLt1NCRaaKhOktHUg9pSibbQS7N0ZhrpbamiqR+PF3ZlBeWr2Vc0saUum6621uoGrcdzV0ZljZ3sF2mmaeWdzFp+12YUdtOV7aburST5pZmMnXjmNBYR3XaCdV1UNMImSoymQwJSf6tSBKWNneSbX6Dzs4OVnckTJs0gTHVaf7n2r2S1SuXs6S7iV0m1VLX3UJ30wxyNQ2kKVSvfJ7u8TuzogOqmhezOm1guwZo6F5NV+N0WnM1vLl0MdtnVlAzbhot7d3Uti6hYcd9yNVPIFnxLE88+zLtTTuy584zqW5+lTdaszRmOpk4fiIdmXqq21extnEGbzR3stO4KrrbVlLTNJmVrZ1U59pZtqqVXWdMpKqrhedXZtl+u2k0dr1JSoY0l6OrYTJJdwddLStoqRpPTfMiJkyZwbK10N6ZpY0akup6xncspqpzDd11E6kfM5Yxb/yBzPR3sLwdtps6jZYuaFm1nMljG3h2WSt7jOvi1Y5G3lLXwpj2Jayd9k5yVXUkaUrvr940B6SkpPn9uS5qVr1E1/i3kmaqe49nOluoW/0idLdDktAxaQ+yNQ2FcdRpb3M9w/V7/zyR/y/T1UYm20FH01tIcp0kaY7q9jepal9Jx4S3kZKS6W7nxVcWsTpbx7QZOzBjQj0rWjr54ysr2Wv78WTTlMmNtXTnciQktHV2U1udoaW9m8lNdYypydCVzf/V0tGd0tmdpbGumvqaDGvWdlFTlaG2KsOK1k66cynTx9cx6F9l640O29xhhStbO8hkEhprq0kSaO/K8vrqdla2djKpsYaZExvoyubo7M7RVF9NVWFsX9+nS9fbkRamP/SrNy3sT6qApLD+QX6YNoX/aptfJUm7yXSvpaNxB7prx/NGcycNddWMyXTT1bqaaY0ZqKqhu25ifth+UpVfo6HndpqDNEt1VyukObK16/+DaoCfy8Z+jxZ21za/Sq66kWz9BOpWPkt3/WTS6jqSbCe52nEk3W1k68Zv9GdcrNGQg72nL738IgBv3fmtQ26/K5tjWUsHNZkM9TUZqos05Hub2sQoyWKMoRzWkZglHOVZjGde9vwTpEkV03bavTzWHimfQbNbaOOFF+sSKdaPJtdnKlu5eO2110hqGzjypFOpq60rdTkDijHS0tKy0ePlFqh7Kh0zyDn1he1gU0g31c7mtCEV12b8ZZVk+s95rK3u/w+kqr5NJAn1tRnqa2vW7apKqK1qyN/J1FE7Lv+LKWHD4RhV1XVsV8+m1U2i55+4UycW/hE6dgJsN9A8l4kUnp1xU3r2wOtLVwIwaeJEmDhxM550nUkzNtw3Gdi5cHtrZ9uMnwzwlo0e34518943lP8YaCYA+27knHdu/Ml33o3t+x3el103cuqe690f6J/hO278mYZw7hEA7DLAkbcNUsOmvXtIjyqGVZmFTKT/B4zvOqhk5RRNOX0YWU615B1V6gIGtCLNf9i9835b92H3bsUoRqPWwtqpAOxV9p0uqmS5QudfuYbpzVFuH2euIR+Gp4QQNgj7hX1TgPYY46pB2ulZt2n6Ro73/BP99aEWKkmSJEka3coqUBdW7X4KqGLgTqFAvuY/b6KpntW91+/U6VlJfA8gW3guSZIkSZK2WFkF6oJ7CtsPDnCsZ9/dW9HGwcBU4CG/g1qSJEmSNFTlGKi/B7QDnwkh9E7qCCHsD3ya/Ard1/XZv2sIYY8QQt8VRh4E/gIcFUL4+z7nTu3z2K8P30uQJEmSJFW6sgvUMcaXgAuAccD8EMIvQwj3AI8AY4FzYoxL+zzkfuBp4EN92sgBHyc/H/uGEMLvQgi3AxGYBXwnxnjXtng9kiRJkqTKVHaBGiDGeB1wIvA74D3AAcBDwFExxps2s43fA3OA28gvhHk08DLwD8C5w1C2JEmSJGkUKbevzeoVY/wF8IvNOG/nQY49BXy4iGVJkiRJkgSUaQ+1JEmSJEnlLknTtNQ1jFgLFy5cBOyQyWRoaGgodTkDamlpAaCpqanElWg08zpUOfA6VLnwWlQ58DpUORgJ12FbWxu5XA7gtdmzZ89c/3jZDvkeIZoAcrlc78VQrsq9Po0OXocqB16HKhdeiyoHXocqByPkOhww9Ruot86LwFvJryb+XIlrkSRJkiQV19vIh+kXBzrokG9JkiRJkobARckkSZIkSRoCA7UkSZIkSUNgoJYkSZIkaQgM1JIkSZIkDYGBWpIkSZKkITBQS5IkSZI0BAZqSZIkSZKGwEAtSZIkSdIQGKglSZIkSRoCA7UkSZIkSUNgoJYkSZIkaQgM1JIkSZIkDYGBWpIkSZKkITBQS5IkSZI0BNWlLkDDI4RwJPA5YBZQCywErogx/qqkhanshRCqgHOBM4C3A1XAC8AtwNdijO3rnb8/cAlwANAE/AW4Osb4k420vztwKfBuYDLwHHADcF2MMTfA+dsX2j8KmAG8AtwEXBlj7Nja16uRIYQwCXgSmBFjTAY4PqzXVQhhAnAR8CFgR+AN4Dbg0hjjmmK8RpWvEMJOwBeAY4BpwDLgf4AvxBiXrHeu16KGTQjhNOA8YG/yHWMR+B5wbYwxu965XosqihDCmeSvs/fEGB8a4HhZXWshhDHAPwOnAW8FVrHud/brW/jyNylJ07TYbarE+lz0HcAD5APR4UAN8IkY4w2lq07lrBCm/xt4P9ACPAp0AQcCEwr33xdjbCucfxT5X1AZ4EGgDTgCGAN8Jcb4+fXa3weYB4wDHgaWkr82JwA/jjGett75M4H5wEzgMfLB/hBgOvAb4OgYY1cxfwYqTyGEW4C/AVg/UA/3dRVCGAf8lvwHlBH4MzCb/F/STwEHxxhXF/UFq2wUPjS8DxhP/r1/Htif/PXzPHBAjHFl4VyvRQ2bEMKVwIXk/303D+gG3kP+w+w7gZNjjGnhXK9FFUUI4SDg1+Svsw0CdbldayGEGvL/Nj0KeBX4PbAH8A5gCTAnxvjKVv5Y+nHId4UJIcwArgdWA/vHGI+PMR5D/kJdA1wdQtihlDWqrJ1NPkz/CdgjxnhkjPE4YDfyv/zmAP8GvZ/+3VR43FExxiNijCeS/9R8EfC5EMLsnoZDCAnwQ/K/cOfGGN8dYzwZ2L3wfP8nhHDKevVcR/4X7r/FGN8ZY/ww8Dby/7h9L/DJYv8AVH5CCKdSCNMDHNsW19W/k/+L/DvAnjHGjxTa/xGwZ+G4KlAIoQ74Cfkw/ckY46wY44fI/068DdgV+GLhXK9FDZsQwt7Av5IfHTErxnh0jPF48kHhJeCDwMmFc70WVRQhhJOBX5EP0wMdL8dr7XzyYfp/gN1ijB+OMe4FfIV8aP/2Fv0QNoOBuvKcD9QB34wxPtmzM8a4ALgSqAfOKVFtKn9nFrb/HGN8rWdnjHE5+WHgAB8rbOeSH/r44xjj//Y593ngs4W7fX8pHkX+F+JvYow39Tl/GfCP658fQgjACeR7gL7S5/xW4CwgS/56VwUrDAu7FniE/Hu+vmG9rgrDzM4m/4HkBT1D12KM3YX2VwJnhRAat/a1qix9lHx4/nGM8ZqenYWpL/9CfthhKOz2WtRwOgpIgJtijM/07Cz8XX1d4e6hfc71WtSQhRBmhhB+SP6Dwyryv+sGUlbXWiHgfwpIgfPWGz7+b+R7uE8IIbx1S34em2KgrjzHFrZ3DnDsjsL2uG1Ui0ae5cBfyQ+PWV/PX+DbF7aDXWt3kf+l2Pda2+j5McaeIULvDiGMLew+hvw/Hu5af/5NYajOH4GdQgh7DvaCNOJ9l/wHgWds5PhwX1eHkp/C8ECMsXm98///9u49Rq6yjOP4d6E0gBUL0iAgaCr6iECiBcpFK1WLGpUqJpCoCUEiBvhH/EOrMTGkKhKNxJiACgWUYIwaQkIQhIhBRASiNSgCj5fK3QBaboXWgqx/vO9pt9PZ2eUsszOd/X6SzcnMeXd7Nv3tmXnmvW2gfKK+G3DcS/y9tGNoelbO7zyRmQ9m5msys8mgWVQ/NRnpNspw73pcX49mUTP1VUrHye8p0/7unaTdsGXtMMrfyJ2ZeV9H+xeBq+vDl7UWsqAeIfVTmbdQbrr3dGny13rukNpW2kZmnpCZB9dPCjsdWY8P1eMh9XhXZ8O6QMQjwKKI2Geq9s23Ue5JzU10qvbNzf2wSc5rBxcRZ1JerFdl5t8nadbvXJnDuW0JsBm4MyIOiIhVEXFxRJwXEUd2tDWL6qfrKb1uJ0XEFyJiUUQsjIjTgM9QeuourW3NombqXsoH2Udl5p97tBu2rA0kmxbUo2VPynDv/2Tm5s6TdXjEv4HdgVd2npcmUz+AWV0fXlmP+9bjZKslNs83BXW/22uERMQbgG9SFlbsNd/JHKov6vzpAyg9LCdR3oidRxl+uAq4oy4S1TCL6pvMvIcyZW8j8HVKLp+gjOL5A3B0Zj5Ym5tFzUhmnpeZl3f2IncxbFkbSDYtqEdLM1fluR5tNtZj18UFpEmcSxlO8yilyIGtedvY9Tu2z9pU+Zxpe42Iutr85ZQRNZ9sVq2dRL9zZQ7nrj3qcS9KHq+izJfek7KWxHrgcxHRrEtiFtVvt1CGuD5L+bDxl8AzwFLgrAmjD82iZsuwZW0g2XQf6tHSfIrU683nWMdR6ikiVlMWGfsvcHJdaALKHOmxHsVOZ9amyudM22t0fB44FvjUNLa26HeuzOHctWs97g7c0LH1y08iYgNwDfDliLgYs6g+ioijKVsX3Q8c2swPrQs3XkUZ9v00Zb90s6jZMmxZG0g27aEeLRvqcbcebZo3CN3myEpbRMS8iPg+ZVXETcCJmXnzhCbPAmMRsWvXH7B91qbK50zbawTU/SzPAa7NzEum8S39zpU5nLsm/p9e2HkyM38OPExZAOcgzKL669uU6XqnTVxsKTMfAT5G2ZP6sxGxO2ZRs2fYsjaQbNpDPVqepgRp74iYV+dMbxER8ygrQW7KzCcHcYHaMUTEAuBnlAWhngQ+3FFMQ1l0bCFlT7/7uvyYznksjwBvre27rRbZrT21fTdTzZPRjulrwHxgl4i4ouPcTgATnj+b/ufKHM5dT1EWJJtP93sclN7C/SmvrWZRfRERu1GGdT9Vt0HdRmaui4ikLMh0EGZRs2fYsjaQbNpDPULq0Nu7KfvFvalLk6D8n/darU9zXETsCdxEKaYfBJZ1KaZh6wqK221bFRF7ULbXejwzH51G+zHgzZRh5HdP1b46uB7N82hp5jUdD3yi46sZotU8XkD/c2UO56jM/B9bd8zYb5JmzZu2xzGL6p9XUe5/L/Ro05ybj1nU7Bm2rA0kmxbUo+cX9fiRLuea566dpWvRDiYi5lPycTjl5ndsZk629UCvrJ1A+WBnRM/9LAAABL5JREFUYtZ6tT8WWATcMmGfwab9yojY5l4VEQcCbwPuz8y70cjIzOWZOdbti/KizITn7qP/ubqZsojJioh4RUf7BcAKysig38zg19bwuq4eT+48EREBvJ7SI7IOs6j+eYyyCN6rI2Jp58mI2J9SKGym9BKaRc2WocpaXQ3/fmBJRBzQ0X4nYCVlfvX1L/H37MmCevRcRpnvuioiDm+ejIgjKAv9bKTLXDCpWg0cTemZXp6ZD/VoeyXlRf7UiPhA82RELKZsLTMOnD+h/a+BvwDHR8TpE9ovYmsmv9U8n5n/pNx4g61bdlFvqGsoBfuW9pqz+pqruif7DykrO19Yp840U2guoEx7uGjCmwWNlu9R5tqdEhEfb56sI3nWUN5HXVC3ljGL6ouarzX14ZpaQAMQEXsDV1B6pi/NzA2YRc2eYczad+vPuaSjCP8KZQTvVZn5j5n92tsaGx/vtSC0dkQRcRYlZM8DN1KGCb2bMmf+lMzsnJcoERF7AQ9RFnJYy9ahjttpVruNiJWUwnpnyk31GeA9lFVxv5SZ53b8G0spmVwA3E7p2VlOuVFenJmf7mi/GPgtZVjlXUBSPvHcl9JztLJzrQCNroh4Adi59lZPfL6vuap/G7dS3gCso/x9LAEWA38E3lnfxGoERcTJwI8or6FrKQuRHUOZN/0r4P2Z+XxtaxbVF3UB0OsoedpEec0dp3wIvhC4DVhRCxCzqJdVRNxE2T51WWbe0nFuqLJWR1veCLyDMle6+d5DgQcoe7Y7h1q9ZeaFlCG3twHLgCMpexcebzGtHpaydVXEJWw/f3XiFwCZeTXlBnsDZZjOccCfKNtrbVNM1/Z3AEdRivA3Au+lDM05AzizS/t19bp+QBk29EHgCeCLwEctpgX9z1Vmrqe82H8H2IVyf30R+AbwLt80jrbM/CnldfRK4EDK/P7HKNsJbimma1uzqL7IzE2UPJ1N6RFcRilaHqBkcXlTTNf2ZlGzYtiylpmbgfdReqSfAz5EKfYvog/FNNhDLUmSJElSK/ZQS5IkSZLUggW1JEmSJEktWFBLkiRJktSCBbUkSZIkSS1YUEuSJEmS1IIFtSRJkiRJLVhQS5IkSZLUggW1JEmSJEktWFBLkiRJktSCBbUkSZIkSS1YUEuSJEmS1IIFtSRJkiRJLcwb9AVIkqThFRGnApcBFwDn1K+VwD7AY8A1wOrM/NdgrlCSpMGxh1qSJE3HfsBa4CzgBeBvwGuBM4DfRcTCAV6bJEkDYUEtSZKm40RgI3BEZi7OzEOBtwPPAa8DTh/kxUmSNAgW1JIkabpOycy1zYPMvBX4cX14zGAuSZKkwbGgliRJ07E+M2/v8nzW4x6zeTGSJA0DC2pJkjQdD0/y/MZ63Hm2LkSSpGFhQS1JkqZj8xTnx2blKiRJGiIW1JIkSZIktWBBLUmSJElSCxbUkiRJkiS1YEEtSZIkSVILFtSSJEmSJLVgQS1JkiRJUgtj4+Pjg74GSZIkSZJ2OPZQS5IkSZLUggW1JEmSJEktWFBLkiRJktSCBbUkSZIkSS1YUEuSJEmS1IIFtSRJkiRJLVhQS5IkSZLUggW1JEmSJEktWFBLkiRJktSCBbUkSZIkSS38H4uFbGxVxqUsAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 1152x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax = plt.subplots(figsize = (16,5))\n",
    "ax = sns.lineplot(x = 'n',\n",
    "                   y = 'p',\n",
    "                   hue = 'fixed',\n",
    "                   data = results,\n",
    "                   ci = 'sd',\n",
    "                   seed = 12345,\n",
    "                   alpha = .7,\n",
    "                   ax = ax,\n",
    "                   )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}