dungeons_soln.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Think Bayes\n",
    "\n",
    "This notebook presents example code and exercise solutions for Think Bayes.\n",
    "\n",
    "Copyright 2018 Allen B. Downey\n",
    "\n",
    "MIT License: https://opensource.org/licenses/MIT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Configure Jupyter so figures appear in the notebook\n",
    "%matplotlib inline\n",
    "\n",
    "# Configure Jupyter to display the assigned value after an assignment\n",
    "%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'\n",
    "\n",
    "# import classes from thinkbayes2\n",
    "from thinkbayes2 import Pmf, Suite\n",
    "\n",
    "import thinkbayes2\n",
    "import thinkplot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Dungeons and Dragons club\n",
    "\n",
    "Suppose there are 10 people in my *Dungeons and Dragons* club; on any game day, each of them has a 70% chance of showing up.\n",
    "\n",
    "Each player has one character and each character has 6 attributes, each of which is generated by rolling and adding up 3 6-sided dice.\n",
    "\n",
    "At the beginning of the game, I ask whose character has the lowest attribute. The wizard says, \"My constitution is 5; does anyone have a lower attribute?\", and no one does.\n",
    "\n",
    "The warrior says \"My strength is 16; does anyone have a higher attribute?\", and no one does.\n",
    "\n",
    "How many characters are in the party?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The prior\n",
    "\n",
    "Since there are 10 players, each with the 70% chance of attending, the prior distribution is [binomial](https://en.wikipedia.org/wiki/Binomial_distribution) with `n=10` and `p=0.7`.\n",
    "\n",
    "The PMF of the binomial distribution is:\n",
    "\n",
    "$ PMF(k; n, p) = P(k ~|~ n, p) = {n \\choose k}\\,p^{k}(1-p)^{n-k}$\n",
    "\n",
    "We could evalate the right hand side in Python, or use `MakeBinomialPmf`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function MakeBinomialPmf in module thinkbayes2.thinkbayes2:\n",
      "\n",
      "MakeBinomialPmf(n, p)\n",
      "    Evaluates the binomial PMF.\n",
      "    \n",
      "    n: number of trials\n",
      "    p: probability of success on each trial\n",
      "    \n",
      "    Returns: Pmf of number of successes\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(thinkbayes2.MakeBinomialPmf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can confirm that the analytic result matches what we computed by convolution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prior = thinkbayes2.MakeBinomialPmf(10, 0.7)\n",
    "thinkplot.Hist(prior)\n",
    "\n",
    "thinkplot.decorate(title='Prior distribution',\n",
    "                   xlabel='Number of players',\n",
    "                   ylabel='PMF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sneaky update\n",
    "\n",
    "Since two players spoke, we can eliminate the possibility of 0 or 1 players, and then renormalize, which is a sneaky version of a Bayesian update.\n",
    "\n",
    "It doesn't make much difference, though, since the prior probabilities for 0 and 1 were so low."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "thinkplot.Pmf(prior, color='gray')\n",
    "del prior[0]\n",
    "del prior[1]\n",
    "prior.Normalize()\n",
    "thinkplot.Pmf(prior)\n",
    "\n",
    "thinkplot.decorate(xlabel='Number of players',\n",
    "                   ylabel='PMF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Likelihood\n",
    "\n",
    "There are three components of the likelihood function:\n",
    "\n",
    "* The probability that the highest attribute is 16.\n",
    "\n",
    "* The probability that the lowest attribute is 5.\n",
    "\n",
    "* The probability that the lowest and highest attributes are held by different players.\n",
    "\n",
    "To compute the first component, we have to compute the distribution of the maximum of $6n$ attributes, where $n$ is the number of players.\n",
    "\n",
    "Here is the distribution for a single die."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 0.16666666666666666\n",
      "2 0.16666666666666666\n",
      "3 0.16666666666666666\n",
      "4 0.16666666666666666\n",
      "5 0.16666666666666666\n",
      "6 0.16666666666666666\n"
     ]
    }
   ],
   "source": [
    "d6 = Pmf([1,2,3,4,5,6])\n",
    "d6.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's the distribution for the sum of three dice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "thrice = sum([d6] * 3)\n",
    "thinkplot.Pdf(thrice)\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='PMF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the CDF for the sum of three dice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cdf_thrice = thrice.MakeCdf()\n",
    "thinkplot.Cdf(cdf_thrice)\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='CDF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Distribution of the maximum\n",
    "\n",
    "The `Max` method raises the CDF to a power."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on method Max in module thinkbayes2.thinkbayes2:\n",
      "\n",
      "Max(k) method of thinkbayes2.thinkbayes2.Cdf instance\n",
      "    Computes the CDF of the maximum of k selections from this dist.\n",
      "    \n",
      "    k: int\n",
      "    \n",
      "    returns: new Cdf\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(cdf_thrice.Max)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So here's the CDF for the maximum of six attributes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cdf_max_6 = cdf_thrice.Max(6)\n",
    "thinkplot.Cdf(cdf_max_6)\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='CDF',\n",
    "                   title='Maximum of 6 attributes')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If there are `n` players, there are `6*n` attributes.  Here are the distributions for the maximum attribute of `n` players, for a few values of `n`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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3oEw/qBsRc0gufihcd1nB4yeBQ7KsIe82btzImWeeySOPPEJHRwef+MQnuPjiiytdlplZxdXcnSSqzR133MH69et5/PHHWbNmDXvvvTdTpkyhpaWl0qWZmVWUp9sokf5OtyGJ1atX09HRwdq1a2loaGDYsGFFn2dmVutqrgeV5YfSil0N05/pNk466STuvvtudtllF9asWcO3v/1tdtppp6xegplZ1ai5gKqk/ky38cc//pH6+nqWL1/Oq6++yqGHHspRRx3F2LFjy1W2mVkuOaBKqD/TbfzsZz9j0qRJDBw4kJ133plDDjmE1tZWB5SZveXVXEDl7UNpxabbGD16NPPmzWPq1KmsWbOG+fPnc95555WxQjOzfPJFEhV2zjnn8Prrr7Pvvvty4IEHcvrppzNu3LhKl2VmVnE114OqlP5OtzFkyBDuuOOOrMoyM6ta7kGZmVkuOaDMzCyXaiagavkWfrX82szMelITAdXY2MjKlStr8hd5RLBy5UoaGxsrXYqZWVnVxEUSzc3NtLW1Uauz7TY2NtLc3FzpMszMyqomAmrgwIGMGTOm0mWYmVkJ1cQQn5mZ1R4HlJmZ5ZIDyszMcskBZWZmueSAMjOzXHJAmZlZLjmgzMwslxxQZmaWSw4oMzPLJQeUmZnlkgPKzMxyqSbuxWdmVm1WsR2v1g1mE+LMG1pL2vbY0cNK2l6luAdlZlYBneGUqYybz5oDysysAsoRToMGVPeveA/xmZlV2A1nTixpe9c/9FxJ26uU6o5XMzOrWQ4oMzPLpUwDStIkSc9IWiLpyz3sc7KkJyUtkvSzLOsxM7Pqkdk5KEn1wHeBo4E2YIGk2RHxZME+uwMXA4dExKuSds6qHjMzqy5Z9qAOApZExNKI2ADcCkzuss9ZwHcj4lWAiFiRYT1mZlZFsgyoXYEXC5bb0nWF9gD2kPR7SfMlTequIUlnS2qV1Nre3p5RuWZmlidZBlR3F/lHl+UBwO7A4cAU4AZJO2zxpIiZETExIiaOGDGi5IWamVn+ZBlQbcCoguVmYHk3+9wdERsj4jngGZLAMjOzt7gsA2oBsLukMZIagFOB2V32uQs4AkBSE8mQ39IMazIzsyqRWUBFRAdwLnAf8BRwe0QsknSFpBPT3e4DVkp6EngAuDAiVmZVk5mZVY9Mb3UUEXOAOV3WXVbwOIDz0y8zM7PNfC8+M7Ne3D3vMW67t5X1GzaWtuE6X/BVjG91ZGbWi0zCqUCdqnxOjAw5oMzMepF1OI3ceYtP1ljKQ3xmZn00a8anS9ZWqWfRrUXuQZmZWS45oMzMLJccUGZmlku9BpSkGwsen5Z5NWZmZqliPaj9Ch5/IctCzMzMChULqK53HzczMyuLYpeZN0u6jmTqjM7Hm0XE5zOrzMzM3tKKBdSFBY990b6ZWYmsfmMTr3dsIoBL7nm6pG3vssOgkrZXKb0GVET8pFyFmJm9lXSGU5bqqvw67aLlSzpN0iOSVqdfrZI+UY7izMxqVTnCqWlwQ8ZHyVavPag0iM4jmQ7jEZJzUQcA10giIm7KvkQzs9p21Ql7lrS9Xy1eUdL2KqVYD+qzwEci4oGIeC0iVkXEPOCj6TYzM7NMFAuoYRGxrOvKdN2wLAoyMzOD4gG1tp/bzMzMtkmxy8z3krSwm/UCxmZQj5mZGVA8oPYD3g682GX9bsDyTCoyMzOj+BDft4G/R8TzhV/AmnSbmZlZJooFVEtEbDHEFxGtQEsmFZmZmVE8oBp72bZdKQsxMzMrVCygFkg6q+tKSWcAf8qmJDMzs+IXSZwH/ELSx/lnIE0EGoCPZFmYmZm9tRW7WexfgfdJOgLYN1393+ndJMzMzDJTrAcFQEQ8ADyQcS1mZmabVfnN2M3MrFY5oMzMLJccUGZmlksOKDMzy6VMA0rSJEnPSFoi6cu97HeSpJA0Mct6zMysevTpKr7+kFQPfBc4Gmgj+dDv7Ih4sst+Q4HPAw9nVYuZWd6M3LGRUcO3o16qmRlwSy3LHtRBwJKIWBoRG4Bbgcnd7HclcDWwLsNazMxypTOcslRfl237WcsyoHblzdN0tKXrNpO0PzAqIn7ZW0OSzpbUKqm1vb299JWamZVZOcJp7PDBmR4ja5kN8ZFMathVbN4o1ZFM2fHJYg1FxExgJsDEiROjyO5mZlXlqD12rnQJuZRlD6oNGFWw3MybJzkcSnL7pN9IWga8F5jtCyXMzAyyDagFwO6SxkhqAE4FZndujIjXIqIpIloiogWYD5yYzjVlZmZvcZkFVER0AOcC9wFPAbdHxCJJV0g6MavjmplZbcjyHBQRMQeY02XdZT3se3iWtZiZWXXxnSTMzCyXHFBmZpZLDigzM8slB5SZmeWSA8rMzHLJAWVmZrnkgDIzs1xyQJmZWS45oMzMLJcyvZOEmVm1W8V2vFo3mE2IM28o3a1Cx44eVrK2apV7UGZmvegMp8xU95yCmXIPysxqwt3zHuO2e1tZv2FjSdvdVDeipO29iWDQAPcTeuKAMrOakEU4FaqTuOHM0k1Xd/1Dz5WsrVrl6DazmpB1OI3ceYfM2rfuuQdlZjVn1oxPl6ytUl4YYVvHPSgzM8slB5SZmeWSA8rMzHLJAWVmZrnkgDIzs1xyQJmZWS45oMzMLJccUGZmlksOKDMzyyUHlJmZ5ZIDyszMcskBZWZmueSAMjOzXHJAmZlZLjmgzMwslxxQZmaWS5kGlKRJkp6RtETSl7vZfr6kJyUtlPRrSbtlWY+ZmVWPzAJKUj3wXeBYYG9giqS9u+z2KDAxIsYBdwJXZ1WPmZlVlyx7UAcBSyJiaURsAG4FJhfuEBEPRMSadHE+0JxhPWZmVkWyDKhdgRcLltvSdT05A7i3uw2SzpbUKqm1vb29hCWamVleZRlQ6mZddLujNBWYCFzT3faImBkREyNi4ogRI0pYopmZ5dWADNtuA0YVLDcDy7vuJOko4BLgsIhYn2E9ZmZWRbIMqAXA7pLGAC8BpwIfK9xB0v7A/wMmRcSKDGsxM+uX1W9s4vWOTQRwyT1Pl6zdXXYYVLK2alVmQ3wR0QGcC9wHPAXcHhGLJF0h6cR0t2uAIcAdkv4saXZW9ZiZ9UdnOGWlzp9G7VGWPSgiYg4wp8u6ywoeH5Xl8c3MtlXW4dQ0uCHDI1S3TAPKzKyWXHXCniVr61eLfVajGHcuzcwslxxQZmaWSw4oMzPLJQeUmZnlkgPKzMxyyQFlZma55IAyM7NcckCZmVku+YO6ZlY2d897jNvubWX9ho2VLsWqgHtQZlY25QinQQ0DM23fyscBZWZlU45wOuXYiZkew8rHQ3xmVhGzZny60iX0ycgdGxk1fDvqJd8/r8wcUGZWE+5b+BdmP7qc9Rs3lbTd3UYNpV7dTRBeGvV12bVd7TzEZ2Y1IYtwAv4ZThnkSH2dGDt8cOkbrhHuQZlZTcginDYTDBpQx1F77JzdMWwLDigzqzk3nFm6CyWuf+i5krVlW8dDfGZmlksOKDMzyyUHlJmZ5ZIDyszMcskBZWZmueSAMjOzXHJAmZlZLjmgzMwslxxQZmaWSw4oMzPLJd/qyMy2UI0z365+YxOvd2wigEvuebpk7e6yw6CStWVbxz0oM9tC1uGUxay3neGUlTr/tiw7v+VmtoWswymLWW+zDqemwQ0ZHsG64yE+M+tVKWe+7ZxU8J6lG7lnaWvJ2u3qqhP2LFlbnkW3cjINKEmTgBlAPXBDRHyzy/ZBwE3ABGAlcEpELMuyJjOrnKwmFYSCqdnrPDV7rcgsoCTVA98FjgbagAWSZkfEkwW7nQG8GhHvknQq8C3glKxqMqsl1Xghw9/WdWR2rujgUUOprxODBmRz5sJTs5dflj2og4AlEbEUQNKtwGSgMKAmA9PTx3cC10tSRGQ2nPzVOU9l1bRZmTXw7iPel+0hBF+fu7hkzb1nzA6bHw/brvQXSkA2FzN4avbKyDKgdgVeLFhuAw7uaZ+I6JD0GjAceKVwJ0lnA2cDjB49Oqt6zayQoD6rS9cy6ozU1cGIwQ2emr1GZBlQ3X0Ldu0Z9WUfImImMBNg4sSJWV6sY1Zd0hCpU/UMP0nQNKSBMcO3L3nb7unUliwDqg0YVbDcDCzvYZ82SQOAtwF/y7AmLj9uryybNzOzEsnyc1ALgN0ljZHUAJwKzO6yz2zgtPTxScC8LM8/mZlZ9cisB5WeUzoXuI/kMvMfRcQiSVcArRExG/ghcLOkJSQ9p1OzqsfMzKpLpp+Diog5wJwu6y4reLwO+NcsazAzs+rkWx2ZmVkuOaDMzCyXHFBmZpZLDigzM8slVdtV3ZLagecrXUcPmuhyF4wq4JrLwzWXR7XVXG31Qmlq3i0iRhTbqeoCKs8ktUZE6Se6yZBrLg/XXB7VVnO11QvlrdlDfGZmlksOKDMzyyUHVGnNrHQB/eCay8M1l0e11Vxt9UIZa/Y5KDMzyyX3oMzMLJccUGZmlksOqBKSVC/pUUm/rHQtfSFpB0l3Snpa0lOS/qXSNRUjaZqkRZKekHSLpMZK19SVpB9JWiHpiYJ1O0m6X9Kz6b87VrLGQj3Ue036fbFQ0i8k7dBbG+XWXc0F2y6QFJKaKlFbT3qqWdLnJD2Tfl9fXan6utPD98Z4SfMl/VlSq6SDsjq+A6q0vgA8VekitsIM4H8iYk9gP3Jeu6Rdgc8DEyNiX5JpXPI4RcuNwKQu674M/Doidgd+nS7nxY1sWe/9wL4RMQ5YDFxc7qKKuJEta0bSKOBo4IVyF9QHN9KlZklHAJOBcRGxD3BtBerqzY1s+T5fDXw1IsYDl6XLmXBAlYikZuB44IZK19IXkoYB7yeZk4uI2BARqypbVZ8MALZLZ2Deni1naa64iHiQLWeGngz8JH38E+DDZS2qF93VGxFzI6IjXZxPMiN2bvTwHgN8G7gIyN3VXz3U/BngmxGxPt1nRdkL60UPNQcwLH38NjL8GXRAlc53SH4wNlW6kD4aC7QDP06HJW+QNLjSRfUmIl4i+QvzBeBl4LWImFvZqvrs7RHxMkD6784Vrmdr/Btwb6WLKEbSicBLEfFYpWvZCnsAh0p6WNJvJR1Y6YL64DzgGkkvkvw8Zta7dkCVgKQPASsi4k+VrmUrDAAOAL4XEfsDq8nXsNMW0vM2k4ExwEhgsKSpla2qtkm6BOgAflrpWnojaXvgEpIhp2oyANgReC9wIXC7JFW2pKI+A0yLiFHANNJRmCw4oErjEOBEScuAW4EPSPqvypZUVBvQFhEPp8t3kgRWnh0FPBcR7RGxEfg58L4K19RXf5W0C0D6b66Gcroj6TTgQ8DHI/8fmHwnyR8uj6U/h83AI5LeUdGqimsDfh6JP5KMwOTq4o5unEbyswdwB+CLJPIsIi6OiOaIaCE5aT8vInL9l31E/AV4UdK701VHAk9WsKS+eAF4r6Tt078yjyTnF3YUmE3yg036790VrKUoSZOALwEnRsSaStdTTEQ8HhE7R0RL+nPYBhyQfp/n2V3ABwAk7QE0kP+7my8HDksffwB4NqsDDciqYasKnwN+KqkBWAqcXuF6ehURD0u6E3iEZNjpUXJ4qxhJtwCHA02S2oDLgW+SDN+cQRK0/1q5Ct+sh3ovBgYB96cjTvMj4tMVK7KL7mqOiMyGmkqhh/f5R8CP0su4NwCn5am32kPNZwEz0guV1gFnZ3b8HL0XZmZmm3mIz8zMcskBZWZmueSAMjOzXHJAmZlZLjmgzMwslxxQZttI0kfSu2fvmS63SPpYwfbxko7r5fkTJV2XPp4u6YKtPP556Z0UzGqKA8ps200Bfsc/76zeAnysYPt4oNuAkjQgIloj4vPbcPzzSG6ca1ZT/Dkos20gaQjwDHAEMDsi9pQ0H9gLeA64BTgH2A54CfhGum0kSZC9QvJh4wsi4kOSppPctmdXYBRwdUT8QNLhnfukx70eaCW5q/S1aQ2vRMQRkj4IfJXkg7b/C5weEa9n/FaYlZx7UGbb5sMkc2otBv4m6QCSm+4+FBHjI+JbJDcwvS1dvi193gRgckR8rJs2x5FM3fIvwGWSRvZ08Ii4juTWM0ek4dQEXAocFREHkITY+aV5qWbl5VsdmW2bKSRTrUByo+ApwH/34XmzI2JtD9t4hEx/AAABFUlEQVTuTretlfQAyc04+zpX13uBvYHfp7coagD+0MfnmuWKA8qsnyQNJ7lZ5r6SgmSG3wDm9OHpq3vZ1nXcPUjuPVg44tHTVPcC7o+IKX2owSzXPMRn1n8nATdFxG7pXbRHkZx32gQMLdjvH12Wi5ksqTENwMOBBcDzwN6SBkl6G8md3Ltrfz5wiKR3QTJPUnqXbLOq44Ay678pwC+6rJtFcjVfh6THJE0DHiAJlz9LOqUP7f6RZJhwPnBlRCyPiBeB24GFJJMHPlqw/0zgXkkPREQ78EngFkkL0zb27PcrNKsgX8VnZma55B6UmZnlkgPKzMxyyQFlZma55IAyM7NcckCZmVkuOaDMzCyXHFBmZpZL/x86/50uRYQsGAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for n in range(2, 10, 2):\n",
    "    cdf_max = cdf_thrice.Max(n*6)\n",
    "    thinkplot.Cdf(cdf_max, label='n=%s'%n)\n",
    "\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='CDF',\n",
    "                   title='Maximum of 6*n attributes')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To check that, I'll compute the CDF for 7 players, estimate it by simulation, and compare."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 7\n",
    "cdf = cdf_thrice.Max(n*6)\n",
    "thinkplot.Cdf(cdf, label='n=%s'%n)\n",
    "\n",
    "sample_max = [max(cdf_thrice.Sample(42)) for i in range(1000)]\n",
    "thinkplot.Cdf(thinkbayes2.Cdf(sample_max), label='sample')\n",
    "\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='CDF',\n",
    "                   title='Maximum of 6*n attributes')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looks good.\n",
    "\n",
    "### Distribution of the minimum\n",
    "\n",
    "Now, to compute the minimum, I have to write my own function, because `Cdf` doesn't provide a `Min` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_cdf_min(cdf, k):\n",
    "    \"\"\"CDF of the min of k samples from cdf.\n",
    "    \n",
    "    cdf: Cdf object\n",
    "    k: number of samples\n",
    "    \n",
    "    returns: new Cdf object\n",
    "    \"\"\"\n",
    "    cdf_min = cdf.Copy()\n",
    "    cdf_min.ps = 1 - (1 - cdf_min.ps)**k\n",
    "    return cdf_min"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute the CDF of the minimum attribute for `n` players, for several values of `n`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mZpWqXmYu6SxJT0palX81SfpkbxRnZmZ9V5c9qDyILiZbDuNJsvei3g1cL4mIuC19iWZm1hdVew/qfwCnRsSiim1zJJ0G3Ak4oHJvrt7AslXraW2FR158odblmJnVvWpDfDu3CycA8m07pyioXrWFU0pK27yZWalUC6g1PdzX5/RGOA3ZzjNTmVnfUW2I7yBJz3SwXcB+CerZJlxz8oGFtnfuTF/hb2Z9T7WAOhzYE1jcbvu+wJIkFZmZmVF9iO/rwJ8j4pXKL2B1vs/MzCyJagHVEBGbDfFFRBPQkKQiMzMzqgfU4C72bV9kIWZmZpWqBdQ8SZ9uv1HSOcATaUoyMzOrfpHExcAPJX2CvwZSIzAQODVlYWZm1rdVmyz2j8D7JR0HHJJv/veImJO8MjMz69Oq9aAAiIhHgEcS12Kd+MOyP7PkjZW0RnDaRf5MlJn1DZ6aoA60hVMqgwYOSNa2mVlPOaDqQOpwOmNCY7L2zcx6qltDfFYes244v9YlmJn1CvegzMyslJIGlKTxkl6UtEDS5V0cd7qkkOSxJjMzAxIGlKT+wE3ABGAMMFnSmA6O2wn4LPBYqlrMzKz+pOxBHQUsiIiFEbGebAXeSR0cdzVwHbA2YS1mZlZnUgbUCN6+TEdzvm0TSUcAoyLix101JOk8SU2SmpYuXVp8pWZmVjopA6qjFco3XS8tqR/Zkh2fq9ZQRMyIiMaIaBw+fHiBJZqZWVmlDKhmYFTF/ZG8fZHDncimT/pPSYuA9wKzfaGEmZlB2oCaB+wvabSkgcCZwOy2nRHxp4gYFhENEdEAzAVOydeaMjOzPi5ZQEVEC3Ah8BPgeeDuiJgv6SpJp6Q6r5mZbRuSziQREQ8AD7TbdkUnxx6bshYzM6svnknCzMxKyXPxFWRdy0bWtbRCwLkz/TaamdnWcg+qIG3hlFK/1CcwMysRB1RReiGcdmtdlfYkZmYl4iG+BGaeW+xHubyKrpn1Re5BmZlZKTmgzMyslBxQZmZWSg4oMzMrJQeUmZmVkgPKzMxKyQFlZmal5IAyM7NSckCZmVkpOaDMzKyUHFBmZlZKnovPzKxGNmzYQHNzM2vXrq11KUkMHjyYkSNHMmDAgB493gFlZlYjzc3N7LTTTjQ0NCCp1uUUKiJYvnw5zc3NjB49ukdteIjPzKxG1q5dy9ChQ7e5cAKQxNChQ7eqd+iAMjOroW0xnNps7ffmgDIzs1JyQJmZ2RZ7+OGHGTduHIceeijjxo1jzpw5hZ/DF0mYmdkWGzZsGD/60Y/Ye++9efbZZznxxBN57bXXCj2HA8rMrAROu+hbydqedcP5ne5btGgREyZM4G//9m/51a9+xYgRI7j//vvZfvvtu2zziCOO2HT74IMPZu3ataxbt45BgwYVVrcDysysj3v55Ze54447uPnmm/nYxz7GrFmzeP311/ne97632bEf/OAHufHGG9+2bdasWRxxxBGFhhM4oMzM+rzRo0czduxYAMaNG8eiRYuYNm0al156adXHzp8/ny984Qs89NBDhdflgDIzK4GuhuFSq+z59O/fnzVr1nD99ddX7UE1Nzdz6qmnctttt/HOd76z8LocUGZmtplLL720yx7UypUrOemkk7j22ms5+uijk9Tgy8zNzGyLffOb32TBggVcffXVjB07lrFjx/LGG28Ueg73oMzM+rCGhgaeffbZTfc///nPd+tx06ZNY9q0aanKAtyDMjOzkkoaUJLGS3pR0gJJl3ew/xJJz0l6RtJPJe2bsh4zM6sfyQJKUn/gJmACMAaYLGlMu8OeAhoj4jDgXuC6VPWYmVl9Sfke1FHAgohYCCDpTmAS8FzbARHxSMXxc4EpCesB4JUVq1m4fBUbW6PQdlsj2NjaCpH2E+FmZn1FyiG+EcDiivvN+bbOnAM82NEOSedJapLUtHTp0q0qKkU4AZvCKUXbbQYN7NmqlGZm9ShlQHW0EEiHr96SpgCNwPUd7Y+IGRHRGBGNw4cP36qikgVIHk4L//hWkuYHDRzAGRMak7RtZlZGKYf4moFRFfdHAkvaHyTpBOBLwDERsS5hPZs54YA9Cmvryz+Yv+n2ozX8RLiZWW969dVXGTNmDNOnT+/2JerdlbIHNQ/YX9JoSQOBM4HZlQdIOgL4v8ApEVHsJ7zMzCy5qVOnMmHChCRtJ+tBRUSLpAuBnwD9gVsiYr6kq4CmiJhNNqQ3BLgnXxr41Yg4JVVNZmZlde7MpmRtzzy387cHerrcBsB9993Hfvvtx4477lhkuZsknUkiIh4AHmi37YqK2yekPL+ZmVXXk+U2Vq1axVe/+lUefvhhvva1ryWpy1MdmZn1cT1ZbuPKK69k6tSpDBkyJFldDigzsxLoahgutZ4st/HYY49x7733ctlll7Fy5Ur69evH4MGDufDCCwurq88F1JurN7Bs1XpaW+GRF1+odTlmZqVUbbmNRx99dNPt6dOnM2TIkELDCfrgZLFt4ZRKRLoP6pqZ9SV9rgeVOpzWv7Um3QnMzArW0+U2Kk2fPr3Aiv6qzwVUpWtOPrCwtsbOmV/9IDMz67Y+N8RnZmb1wQFlZmal5IAyM7NSckCZmVkpOaDMzKyUHFBmZtYjzzzzDO973/s4+OCDOfTQQ1m7dm2h7fe5y8zXtWxkXUu2+m3K2YPNzLZlLS0tTJkyhdtvv53DDz+c5cuXM2BAsat+98GAau1kXd9i9EvZuJlts770o3RTr3X1mc+eLrfx0EMPcdhhh3H44YcDMHTo0EJrhj4YUKnDabfWVelOYGaWQE+W23jppZeQxIknnsjSpUs588wzueyyywqtq+8FVIUiZw8+7SIPF5pZferJchstLS384he/YN68eeywww4cf/zxjBs3juOPP76wuvp0QJmZlUWRU69tqZ4stzFy5EiOOeYYhg0bBsDEiRN58sknHVBmZpZWteU2TjzxRK677jpWr17NwIED+dnPfsbUqVMLrcEBZWZmW2y33Xbjkksu4cgjj0QSEydO5KSTTir0HA4oM7M+bGuW25gyZQpTpkxJURbgD+qamVlJOaDMzKyU+twQX2sEG1uzD+uedtG3al2OmZl1os/1oNrCKZVBA4ud6sPMrK/qcwGVOpzOmFDch3/NzPqyPjfEV2nWDefXugQzM+tE3+tBmZnZVtuwYQNnnXUWhx56KAcddBDXXntt4efo0z0oMzPrmXvuuYd169bx29/+ltWrVzNmzBgmT55MQ0NDYedwQJmZlcB/vPRGsrZPOGCPTvf1dLkNSaxatYqWlhbWrFnDwIED2XnnnQut2wFlZtbH9WS5jdNPP53777+fvfbai9WrV/P1r3+d3XffvdC6HFBmZn1cT5bbePzxx+nfvz9LlixhxYoVfOADH+CEE05gv/32K6yupAElaTxwA9AfmBkRX2m3fxBwGzAOWA6cERGLUtZkZlZGXQ3DpdaT5Ta+//3vM378eAYMGMAee+zB0UcfTVNTU30ElKT+wE3Ah4FmYJ6k2RHxXMVh5wArIuJdks4EvgqckaomMzPrnmrLbeyzzz7MmTOHKVOmsHr1aubOncvFF19caA0pLzM/ClgQEQsjYj1wJzCp3TGTgO/kt+8FjpekhDWZmVkBLrjgAt566y0OOeQQjjzySM4++2wOO+ywQs+RcohvBLC44n4z8J7OjomIFkl/AoYCyyoPknQecB5kqW1mZsXo6XIbQ4YM4Z577klVFpC2B9VRT6j9REPdOYaImBERjRHROHz48EKKMzOzckvZg2oGRlXcHwks6eSYZknbAbsAbyasiSsnHpSyeTMzK0jKHtQ8YH9JoyUNBM4EZrc7ZjZwVn77dGBORCScztXMrFy25Ze8rf3ekgVURLQAFwI/AZ4H7o6I+ZKuknRKfti3gaGSFgCXAJenqsfMrGwGDx7M8uXLt8mQigiWL1/O4MGDe9yG6u2JaWxsjKamplqXYWa21TZs2EBzczNr166tdSlJDB48mJEjRzJgwNvXyZP0RERUXZvIM0mYmdXIgAEDGD16dK3LKC0vt2FmZqXkgDIzs1JyQJmZWSnV3UUSkpYCr9S6jk4Mo90sGHXANfcO19w76q3meqsXiql534ioOutC3QVUmUlq6s6VKWXimnuHa+4d9VZzvdULvVuzh/jMzKyUHFBmZlZKDqhizah1AT3gmnuHa+4d9VZzvdULvViz34MyM7NScg/KzMxKyQFlZmal5IAqkKT+kp6S9ONa19IdknaVdK+kFyQ9L+l9ta6pGklTJc2X9KykOyT1fKrkRCTdIukNSc9WbNtd0sOSXs7/3a2WNVbqpN7r85+LZyT9UNKutayxvY5qrtj3eUkhaVgtautMZzVL+oykF/Of6+tqVV9HOvnZGCtprqTfSGqSdFSq8zuginUR2dIi9eIG4P9FxIHA4ZS8dkkjgM8CjRFxCNCfbJ2xsrkVGN9u2+XATyNif+CnlGtpmVvZvN6HgUMi4jDgJeCLvV1UFbeyec1IGgV8GHi1twvqhltpV7Ok44BJwGERcTDwtRrU1ZVb2fx5vg74ckSMBa7I7yfhgCqIpJHAScDMWtfSHZJ2Bj5ItiYXEbE+IlbWtqpu2Q7YPl+BeQc2X6W55iLi52y+MvQk4Dv57e8AH+3VorrQUb0R8VC+phvAXLIVsUujk+cY4OvAZUDprv7qpOZ/AL4SEevyY97o9cK60EnNAeyc396FhL+DDqjifIPsF6O11oV0037AUuDf8mHJmZJ2rHVRXYmI18j+wnwVeB34U0Q8VNuqum3PiHgdIP93jxrXsyX+G/BgrYuoJl8I9bWIeLrWtWyBA4APSHpM0s8kHVnrgrrhYuB6SYvJfh+T9a4dUAWQ9BHgjYh4ota1bIHtgHcD/xoRRwCrKNew02by920mAaOBvYEdJU2pbVXbNklfAlqA79W6lq5I2gH4EtmQUz3ZDtgNeC9wKXC3JNW2pKr+AZgaEaOAqeSjMCk4oIpxNHCKpEXAncCHJH23tiVV1Qw0R8Rj+f17yQKrzE4Afh8RSyNiA/AD4P01rqm7/ihpL4D831IN5XRE0lnAR4BPRPk/MPlOsj9cns5/D0cCT0p6R02rqq4Z+EFkHicbgSnVxR0dOIvsdw/gHsAXSZRZRHwxIkZGRAPZm/ZzIqLUf9lHxB+AxZL+Jt90PPBcDUvqjleB90raIf8r83hKfmFHhdlkv9jk/95fw1qqkjQe+AJwSkSsrnU91UTEbyNij4hoyH8Pm4F35z/nZXYf8CEASQcAAyn/7OZLgGPy2x8CXk51Ii/53rd9BviepIHAQuDsGtfTpYh4TNK9wJNkw05PUcKpYiTdARwLDJPUDFwJfIVs+OYcsqD9L7Wr8O06qfeLwCDg4XzEaW5EnF+zItvpqOaISDbUVIROnudbgFvyy7jXA2eVqbfaSc2fBm7IL1RaC5yX7Pwlei7MzMw28RCfmZmVkgPKzMxKyQFlZmal5IAyM7NSckCZmVkpOaDMtpKkU/PZsw/M7zdI+njF/rGSJnbx+EZJN+a3p0v6/Bae/+J8JgWzbYoDymzrTQZ+wV9nVm8APl6xfyzQYUBJ2i4imiLis1tx/ovJJs4126b4c1BmW0HSEOBF4DhgdkQcKGkucBDwe+AO4AJge+A14Np8395kQbaM7MPGn4+Ij0iaTjZtzwhgFHBdRNws6di2Y/LzfhNoIptV+mt5Dcsi4jhJfwd8meyDtr8Dzo6ItxI/FWaFcw/KbOt8lGxNrZeANyW9m2zS3UcjYmxEfJVsAtO78vt35Y8bB0yKiI930OZhZEu3vA+4QtLenZ08Im4km3rmuDychgHTgBMi4t1kIXZJMd+qWe/yVEdmW2cy2VIrkE0UPBn49248bnZErOlk3/35vjWSHiGbjLO7a3W9FxgD/DKfomgg8OtuPtasVBxQZj0kaSjZZJmHSAqyFX4DeKAbD1/Vxb724+5BNvdg5YhHZ0vdC3g4IiZ3owazUvMQn1nPnQ7cFhH75rNojyJ736kV2KniuL+0u1/NJEmD8wA8FpgHvAKMkTRI0i5kM7l31P5c4GhJ74JsnaR8lmyzuuOAMuu5ycAP222bRXY1X4ukpyVNBR4hC5ffSDqjG+0+TjZMOBe4OiKWRMRi4G7gGbL/6XGHAAAAU0lEQVTFA5+qOH4G8KCkRyJiKfAp4A5Jz+RtHNjj79CshnwVn5mZlZJ7UGZmVkoOKDMzKyUHlJmZlZIDyszMSskBZWZmpeSAMjOzUnJAmZlZKf1/cR9D1564TLUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for n in range(2, 10, 2):\n",
    "    cdf_min = compute_cdf_min(cdf_thrice, n*6)\n",
    "    thinkplot.Cdf(cdf_min, label='n=%s'%n)\n",
    "\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='CDF',\n",
    "                   title='Minimum of 6*n attributes')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And again we can check it by comparing to simulation results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 7\n",
    "cdf = compute_cdf_min(cdf_thrice, n*6)\n",
    "thinkplot.Cdf(cdf, label='n=%s'%n)\n",
    "\n",
    "sample_min = [min(cdf_thrice.Sample(42)) for i in range(1000)]\n",
    "thinkplot.Cdf(thinkbayes2.Cdf(sample_min), label='sample')\n",
    "\n",
    "thinkplot.decorate(xlabel='Attribute',\n",
    "                   ylabel='CDF',\n",
    "                   title='Minimum of 6*n attributes')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For efficiency and conciseness, it is helpful to precompute the distributions for the relevant values of `n`, and store them in dictionaries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.23288163935889017 0.23288163935889017\n",
      "0.28826338107405935 0.2882633810740594\n",
      "0.31794402625472684 0.3179440262547268\n",
      "0.32955796250238156 0.32955796250238156\n",
      "0.32871475364520075 0.3287147536452008\n",
      "0.3195146933518256 0.3195146933518255\n",
      "0.3049352170780888 0.30493521707808885\n",
      "0.2871203018328896 0.28712030183288956\n",
      "0.2675970126720095 0.2675970126720096\n"
     ]
    }
   ],
   "source": [
    "like_min = {}\n",
    "like_max = {}\n",
    "\n",
    "for n in range(2, 11):\n",
    "    cdf_min = compute_cdf_min(cdf_thrice, n*6)\n",
    "    like_min[n] = cdf_min.MakePmf()\n",
    "    cdf_max = cdf_thrice.Max(n*6)\n",
    "    like_max[n] = cdf_max.MakePmf()\n",
    "    print(like_min[n][5], like_max[n][16])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The output shows that the particular data we saw is symmetric: the chance that 16 is the maximum is the same as the chance that 5 is the minimum.\n",
    "\n",
    "### Probability of different players\n",
    "\n",
    "Finally, we need the probability that the minimum and maximum are held by the same person.  If there are `n` players, there are `6*n` attributes.\n",
    "\n",
    "Let's call the player with the highest attribute Max.  What is the chance that Max also has the lowest attribute?  Well Max has 5 more attributes, out of a total of `6*n-1` remaining attributes.\n",
    "\n",
    "So here's `prob_same` as a function of `n`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 0.45454545454545453\n",
      "3 0.29411764705882354\n",
      "4 0.21739130434782608\n",
      "5 0.1724137931034483\n",
      "6 0.14285714285714285\n",
      "7 0.12195121951219512\n",
      "8 0.10638297872340426\n",
      "9 0.09433962264150944\n",
      "10 0.0847457627118644\n"
     ]
    }
   ],
   "source": [
    "def prob_same(n):\n",
    "    return 5 / (6*n-1)\n",
    "\n",
    "for n in range(2, 11):\n",
    "    print(n, prob_same(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The update"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a class that implements a likelihood function that computes all three parts: the chance that the maximum is 16, the minimum is 5, and they are held by different people."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Dungeons(Suite):\n",
    "    \n",
    "    def Likelihood(self, data, hypo):\n",
    "        \"\"\"Probability of the data given the hypothesis.\n",
    "        \n",
    "        data: lowest attribute, highest attribute, boolean\n",
    "              (whether the same person has both)\n",
    "        hypo: number of players\n",
    "        \n",
    "        returns: probability\n",
    "        \"\"\"\n",
    "        lowest, highest, same = data\n",
    "        n = hypo\n",
    "        \n",
    "        p = prob_same(n)\n",
    "        like = p if same else 1-p\n",
    "        \n",
    "        like *= like_min[n][lowest]\n",
    "        like *= like_max[n][highest]\n",
    "\n",
    "        return like"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the prior we computed above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.0008681450402"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "suite = Dungeons(prior)\n",
    "thinkplot.Hist(suite)\n",
    "thinkplot.decorate(title='Prior distribution',\n",
    "                   xlabel='Number of players',\n",
    "                   ylabel='PMF')\n",
    "suite.Mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's the update based on the data in the problem statement."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.08548474490284352"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "suite.Update((5, 16, False))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's the posterior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.940862784521087"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAagAAAEYCAYAAAAJeGK1AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMi4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvhp/UCwAAF/ZJREFUeJzt3X20XXV95/H3xyCgUhGa1FGeEkpUGB2lE1HHjtKCGqsFVxcoTFGc4jBWo1alXagUkNoZH+pDXeAIowilCiJaTV1BpPKgHQUSQHmyaIgIMVRig1oeRALf+ePs4OF4kntDsjm/5L5fa51198Pvt/f3brj3k98++/5OqgpJklrzqEkXIEnSOAaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlNRJcn6SI3s8fiXZq1v+eJK/3EzH3T3JnUlmdeuXJHnd5jh2d7xer4u0PttMugCpL0luBp4I3A/cBSwB3lRVd45rX1UvfaRqq6rXT6dd9z28rqr+aQPHugXYYXPUleREYK+qOmLo+I/YdZGGOYLS1u4Pq2oH4HeAZwPHjTbIwMP+WUgysX/oTfLcUt8MKM0IVfUj4Hzg6fDgbbC/TvL/gLuBPYdvjSV5VJLjkvwwye1J/i7Jjt2+ud3tuqOS3AJcNO6cSf48yW1JViX5k5F9ZyR5T7c8O8mXk/w0yZok3+jOfxawO/CP3S28vxh37qFtw2H120muSPKzJF9KsnN3rv2TrByp5eYkByZZCLwTeFV3vu8MXauNuS5HJrklyU+SvOth/0fTjGdAaUZIshvwB8DVQ5tfDRwN/Abww5Eur+1evwfsyeAW2skjbV4I7A28ZMz5FgLHAC8C5gMHbqC8twMrgTkMbkm+E6iqejVwC90osKreP51zd14D/AnwZGAt8NENnB8GJ/wK8L+Az3bne+aYZq9l6uvyu8BTgQOA45PsPdW5pXEMKG3tvpjkp8A/A5cy+AW8zhlVdX1Vra2q+0b6/THwoapa0b1n9Q7gsJFRyolVdVdV3TPmvK8EPlVV11XVXcCJG6jxPuBJwB5VdV9VfaOmniRzQ+cGOGvo3H8JvHLdQxSbaDrX5d1VdU9VfQf4DjAu6KQpGVDa2r2iqp5QVXtU1RtGfqHfuoF+T+aho6ofMnio6Ikb0X94/+gIbdgHgOXAV5OsSHLsBtpO59yj+38IPBqYPY3jTmU61+Vfh5bvZjM9wKGZx4DSTLahUcoqYI+h9d0Z3Cr78TT73wbsNtJ/fBFV/15Vb6+qPYE/BN6W5IApzjHVCGv03PcBP2HwNONj1+3oRlVzNuK407ku0mZhQEnjnQ28Ncm8JDvwq/dm1k6z/7nAa5Psk+SxwAnra5jk5Un2ShLg5wwei7+/2/1jBu/1bKwjhs59EnBeVd0PfA/YPsnLkjyawVON2w31+zEwdwNPNW7qdZGmzYCSxjsdOAv4OvAD4BfAm6bbuarOBz7C4Am/5aznSb/OfOCfgDuBbwEfq6pLun3/Gziue8LvmI2o/yzgDAa327YH3tzV9TPgDcAngB8xGFENP9X3ue7rvyW5asxxN+m6SBsjfmChJKlFjqAkSU0yoCRJTTKgJElNMqAkSU3aaiaanD17ds2dO3fSZUiSpnDllVf+pKrmTNVuqwmouXPnsmzZskmXIUmaQpINzazyIG/xSZKaZEBJkppkQEmSmmRASZKaZEBJkppkQEmSmmRASZKaZEBJkppkQEmSmrTVzCQhaXoWvefsSZfAyccdPukStAVwBCVJapIBJUlqkgElSWqSASVJapIBJUlqkgElSWqSASVJapIBJUlqkgElSWqSASVJapIBJUlqkgElSWqSASVJapIBJUlqkgElSWqSASVJapIBJUlqkgElSWpSrwGVZGGSG5MsT3LsmP1vS3JDkmuSfC3JHkP77k/y7e61uM86JUnt2aavAyeZBZwCvAhYCSxNsriqbhhqdjWwoKruTvKnwPuBV3X77qmqZ/VVnySpbX2OoPYDllfViqr6JXAOcPBwg6q6uKru7lYvA3btsR5J0hakz4DaBbh1aH1lt219jgLOH1rfPsmyJJclecW4DkmO7tosW7169aZXLElqRm+3+ICM2VZjGyZHAAuAFw5t3r2qViXZE7goybVVddNDDlZ1GnAawIIFC8YeW5K0ZepzBLUS2G1ofVdg1WijJAcC7wIOqqp7122vqlXd1xXAJcC+PdYqSWpMnwG1FJifZF6SbYHDgIc8jZdkX+BUBuF0+9D2nZJs1y3PBp4PDD9cIUnayvV2i6+q1iZZBFwAzAJOr6rrk5wELKuqxcAHgB2AzyUBuKWqDgL2Bk5N8gCDEH3vyNN/kqStXJ/vQVFVS4AlI9uOH1o+cD39vgk8o8/aJEltcyYJSVKTDChJUpMMKElSkwwoSVKTDChJUpMMKElSkwwoSVKTDChJUpMMKElSkwwoSVKTDChJUpN6nYtPkqZj0XvOnnQJnHzc4ZMuQSMcQUmSmmRASZKaZEBJkppkQEmSmmRASZKaZEBJkppkQEmSmuTfQUk98u97pIfPEZQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSb0GVJKFSW5MsjzJsWP2vy3JDUmuSfK1JHsM7Tsyyfe715F91ilJak9vAZVkFnAK8FJgH+DwJPuMNLsaWFBV/wk4D3h/13dn4ATgOcB+wAlJduqrVklSe/ocQe0HLK+qFVX1S+Ac4ODhBlV1cVXd3a1eBuzaLb8EuLCq1lTVHcCFwMIea5UkNabPgNoFuHVofWW3bX2OAs7fmL5Jjk6yLMmy1atXb2K5kqSW9BlQGbOtxjZMjgAWAB/YmL5VdVpVLaiqBXPmzHnYhUqS2tNnQK0Edhta3xVYNdooyYHAu4CDqurejekrSdp69RlQS4H5SeYl2RY4DFg83CDJvsCpDMLp9qFdFwAvTrJT93DEi7ttkqQZorfPg6qqtUkWMQiWWcDpVXV9kpOAZVW1mMEtvR2AzyUBuKWqDqqqNUn+ikHIAZxUVWv6qlWS1J5eP7CwqpYAS0a2HT+0fOAG+p4OnN5fdZKkljmThCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkGlCSpSQaUJKlJBpQkqUkbDKgkZwwtH9l7NZIkdaYaQT1zaPktfRYiSdKwqQKqNuXgSRYmuTHJ8iTHjtn/giRXJVmb5JCRffcn+Xb3WrwpdUiStjzbTLF/1yQfBTK0/KCqevP6OiaZBZwCvAhYCSxNsriqbhhqdgvwWuCYMYe4p6qeNfW3IEnaGk0VUH8+tLxsI4+9H7C8qlYAJDkHOBh4MKCq6uZu3wMbeWxJ0lZugwFVVWduwrF3AW4dWl8JPGcj+m+fZBmwFnhvVX1xtEGSo4GjAXbfffdNKFWS1JoNBtRU7/1U1UEb6j6uy3SK6uxeVauS7AlclOTaqrpp5PynAacBLFiwYJPeL5MktWWqW3zPYzAKOhu4nPGhsz4rgd2G1ncFVk23c1Wt6r6uSHIJsC9w0wY7SZK2GlM9xfcfgHcCTwf+lsEDDz+pqkur6tIp+i4F5ieZl2Rb4DBgWk/jJdkpyXbd8mzg+Qy9dyVJ2vptMKCq6v6q+kpVHQk8F1gOXJLkTVMduKrWAouAC4DvAudW1fVJTkpyEECSZydZCRwKnJrk+q773sCyJN8BLmbwHpQBJUkzyFS3+OhGMi8DDgfmAh8FvjCdg1fVEmDJyLbjh5aXMrj1N9rvm8AzpnMOSdLWaaqHJM5kcHvvfODdVXXdI1KVJGnGm2oE9WrgLuApwFuSrHtSLkBV1eP7LE6SNHNN9XdQznYuSZqIqW7xbQ+8HtgLuAY4vXv4QZKkXk01QjoTWABcC/wB8MHeK5Ikianfg9qnqp4BkOSTwBX9lyRJ0tQjqPvWLXhrT5L0SJpqBPXMJD/vlgM8plv3KT5JUq+meopv1iNViCRJw3yMXJLUJANKktQkA0qS1CQDSpLUJANKktQkA0qS1CQDSpLUJANKktQkA0qS1CQDSpLUJANKktQkA0qS1CQDSpLUJANKktQkA0qS1CQDSpLUpKk+UVfaIix6z9mTLgGAk487fNIlSFsNR1CSpCY5gpKkMVoYlc/0EXmvI6gkC5PcmGR5kmPH7H9BkquSrE1yyMi+I5N8v3sd2WedkqT29BZQSWYBpwAvBfYBDk+yz0izW4DXAp8Z6bszcALwHGA/4IQkO/VVqySpPX2OoPYDllfViqr6JXAOcPBwg6q6uaquAR4Y6fsS4MKqWlNVdwAXAgt7rFWS1Jg+A2oX4Nah9ZXdts3WN8nRSZYlWbZ69eqHXagkqT19BlTGbKvN2beqTquqBVW1YM6cORtVnCSpbX0G1Epgt6H1XYFVj0BfSdJWoM+AWgrMTzIvybbAYcDiafa9AHhxkp26hyNe3G2TJM0QvQVUVa0FFjEIlu8C51bV9UlOSnIQQJJnJ1kJHAqcmuT6ru8a4K8YhNxS4KRumyRphuj1D3WragmwZGTb8UPLSxncvhvX93Tg9D7rkyS1y6mOJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElN6jWgkixMcmOS5UmOHbN/uySf7fZfnmRut31uknuSfLt7fbzPOiVJ7dmmrwMnmQWcArwIWAksTbK4qm4YanYUcEdV7ZXkMOB9wKu6fTdV1bP6qk+S1LY+R1D7AcurakVV/RI4Bzh4pM3BwJnd8nnAAUnSY02SpC1EnwG1C3Dr0PrKbtvYNlW1FvgZ8JvdvnlJrk5yaZL/Ou4ESY5OsizJstWrV2/e6iVJE9VnQI0bCdU029wG7F5V+wJvAz6T5PG/1rDqtKpaUFUL5syZs8kFS5La0WdArQR2G1rfFVi1vjZJtgF2BNZU1b1V9W8AVXUlcBPwlB5rlSQ1ps+AWgrMTzIvybbAYcDikTaLgSO75UOAi6qqkszpHrIgyZ7AfGBFj7VKkhrT21N8VbU2ySLgAmAWcHpVXZ/kJGBZVS0GPgmclWQ5sIZBiAG8ADgpyVrgfuD1VbWmr1olSe3pLaAAqmoJsGRk2/FDy78ADh3T7/PA5/usTZLUNmeSkCQ1yYCSJDXJgJIkNcmAkiQ1yYCSJDXJgJIkNanXx8y1dVr0nrMnXQIAJx93+KRLkNQjR1CSpCYZUJKkJhlQkqQmGVCSpCYZUJKkJhlQkqQmGVCSpCYZUJKkJhlQkqQmOZOEJG0hZtosLo6gJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNMqAkSU0yoCRJTTKgJElNcqqjxs20qU0kaR1HUJKkJvUaUEkWJrkxyfIkx47Zv12Sz3b7L08yd2jfO7rtNyZ5SZ91SpLa01tAJZkFnAK8FNgHODzJPiPNjgLuqKq9gA8D7+v67gMcBvxHYCHwse54kqQZos/3oPYDllfVCoAk5wAHAzcMtTkYOLFbPg84OUm67edU1b3AD5Is7473rR7rbeL9Ht/rkaSBVFU/B04OARZW1eu69VcDz6mqRUNtruvarOzWbwKewyC0Lquqv++2fxI4v6rOGznH0cDR3epTgRt7+WY2zmzgJ5MuYgvgdZoer9P0ea2mp4XrtEdVzZmqUZ8jqIzZNpqG62sznb5U1WnAaRtfWn+SLKuqBZOuo3Vep+nxOk2f12p6tqTr1OdDEiuB3YbWdwVWra9Nkm2AHYE10+wrSdqK9RlQS4H5SeYl2ZbBQw+LR9osBo7slg8BLqrBPcfFwGHdU37zgPnAFT3WKklqTG+3+KpqbZJFwAXALOD0qro+yUnAsqpaDHwSOKt7CGINgxCja3cugwcq1gJvrKr7+6p1M2vqlmPDvE7T43WaPq/V9Gwx16m3hyQkSdoUziQhSWqSASVJapIBtRkk2S3JxUm+m+T6JG+ZdE0tSzIrydVJvjzpWlqW5AlJzkvyL93/W8+bdE0tSvLW7ufuuiRnJ9l+0jW1IsnpSW7v/uZ03badk1yY5Pvd150mWeOGGFCbx1rg7VW1N/Bc4I1jpnXSr7wF+O6ki9gC/C3wlap6GvBMvGa/JskuwJuBBVX1dAYPZB022aqacgaD6eKGHQt8rarmA1/r1ptkQG0GVXVbVV3VLf87g18ku0y2qjYl2RV4GfCJSdfSsiSPB17A4ElXquqXVfXTyVbVrG2Ax3R/S/lY/JvJB1XV1xk8IT3sYODMbvlM4BWPaFEbwYDazLoZ2fcFLp9sJc36CPAXwAOTLqRxewKrgU91t0M/keRxky6qNVX1I+BvgFuA24CfVdVXJ1tV855YVbfB4B/XwG9NuJ71MqA2oyQ7AJ8H/qyqfj7pelqT5OXA7VV15aRr2QJsA/wO8H+qal/gLhq+FTMp3fsnBwPzgCcDj0tyxGSr0uZiQG0mSR7NIJw+XVVfmHQ9jXo+cFCSm4FzgN9P8veTLalZK4GVVbVuJH4eg8DSQx0I/KCqVlfVfcAXgP8y4Zpa9+MkTwLovt4+4XrWy4DaDLqPCPkk8N2q+tCk62lVVb2jqnatqrkM3si+qKr81+4YVfWvwK1JntptOoCHflSNBm4Bnpvksd3P4QH4MMlUhqeYOxL40gRr2aA+ZzOfSZ4PvBq4Nsm3u23vrKolE6xJW743AZ/u5rJcAfz3CdfTnKq6PMl5wFUMnqa9mi1oKp++JTkb2B+YnWQlcALwXuDcJEcxCPhDJ1fhhjnVkSSpSd7ikyQ1yYCSJDXJgJIkNcmAkiQ1yYCSJDXJgNKMkaSSfHBo/ZgkJ26mY5+R5JDNcawpznNoN7P5xRvR5+Yks/usS+qDAaWZ5F7gj1r7ZZ1k1kY0Pwp4Q1X9Xl/1bKyNrF+aNgNKM8laBn/E+dbRHaMjoCR3dl/3T3JpknOTfC/Je5P8cZIrklyb5LeHDnNgkm907V7e9Z+V5ANJlia5Jsn/HDruxUk+A1w7pp7Du+Nfl+R93bbjgd8FPp7kAyPt90/y9ST/kOSGJB9P8ms/30m+mOTK7vOTju62HZXkw0Nt/keSD3XLR3Tf67eTnLoujJLcmeSkJJcDz+uuyw3d9/g30/qvIU2lqnz5mhEv4E7g8cDNwI7AMcCJ3b4zgEOG23Zf9wd+CjwJ2A74EfDubt9bgI8M9f8Kg3/0zWcwl972wNHAcV2b7YBlDCY23Z/BBLDzxtT5ZAZ/4T+HwWwvFwGv6PZdwuCzj0b77A/8gsEs6LOAC9d9P933O7tb3rn7+hjgOuA3gccBNwGP7vZ9E3gGsDfwj0PbPwa8plsu4JXrjgncyK/+8P8Jk/5v7WvreDmC0oxSg1nm/47Bh9xN19IafObXvQx+ka/7OIdrgblD7c6tqgeq6vsMpiZ6GvBi4DXdFFiXMwiE+V37K6rqB2PO92zgkhpMgLoW+DSDz4aayhVVtaKq7gfOZjDaGvXmJN8BLgN2A+ZX1V0MQvDlSZ7GIJCuZTCv3X8Glnb1H8AgAAHuZzA5MsDPGYTjJ5L8EXD3NGqVpuRcfJqJPsJg7rZPDW1bS3fLu5t0dNuhffcOLT8wtP4AD/0ZGp03rIAAb6qqC4Z3JNmfwQhqnEz5HYw37vyj5zwQeF5V3Z3kEgajPBh8gOQ7gX/hV9clwJlV9Y4x5/pFF4RU1dok+zEIsMOARcDvP8zvQXqQIyjNOFW1BjiXwQMH69zMYLQAg88XevTDOPShSR7VvS+1J4PbXhcAf9p9HAtJnjKNDx68HHhhktndez6HA5dO4/z7JZnXvff0KuCfR/bvCNzRhdPTgOeu21GDj/XYDfhvDEZfMPg48EOS/FZX+85J9hg9afc5aDvWYHLkPwOeNY1apSk5gtJM9UEG/9Jf5/8CX0pyBYNfzOsb3WzIjQyC5InA66vqF0k+weA24FXdyGw1U3zEdlXdluQdwMUMRjFLqmo6H4nwLQYzVT8D+DrwDyP7vwK8Psk1Xa2Xjew/F3hWVd3R1XFDkuOAr3ahdx/wRuCHI/1+g8G1276r99ceQpEeDmczl7YC3e27Y6rq5ZtwjC8DH66qr222wqRN4C0+aYZL8oQk3wPuMZzUEkdQkqQmOYKSJDXJgJIkNcmAkiQ1yYCSJDXJgJIkNen/A5h74b1IJWf6AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "thinkplot.Hist(suite)\n",
    "thinkplot.decorate(title='Prior distribution',\n",
    "                   xlabel='Number of players',\n",
    "                   ylabel='PMF')\n",
    "suite.Mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For these data, the posterior is not very different from the prior.  The mean is slightly lower, which means that these data suggest there might be fewer than the usual number of players."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 0.0005007044860801899\n",
      "3 0.006177449626400222\n",
      "4 0.03402191676923564\n",
      "5 0.10823000113979396\n",
      "6 0.21684925990462958\n",
      "7 0.2798371634966231\n",
      "8 0.22697589141459001\n",
      "9 0.10574761295351945\n",
      "10 0.0216600002091279\n"
     ]
    }
   ],
   "source": [
    "suite.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Based on the data, I am 94% sure there are between 5 and 9 players."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 9)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "suite.CredibleInterval()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.937639928909156"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(suite[n] for n in [5,6,7,8,9])"
   ]
  }
 ],
 "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": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}