Create an N-faced fair die from throwing an M-faced one.

die_n_from_m.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#Create-an-N-sided-die-from-an-M-sided-die\" data-toc-modified-id=\"Create-an-N-sided-die-from-an-M-sided-die-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Create an N-sided die from an M-sided die</a></div><div class=\"lev2 toc-item\"><a href=\"#What-does-fair-mean?\" data-toc-modified-id=\"What-does-fair-mean?-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>What does fair mean?</a></div><div class=\"lev1 toc-item\"><a href=\"#A-six-sided-die-simulation\" data-toc-modified-id=\"A-six-sided-die-simulation-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>A six-sided die simulation</a></div><div class=\"lev2 toc-item\"><a href=\"#How-good-is-the-simulation?\" data-toc-modified-id=\"How-good-is-the-simulation?-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>How good is the simulation?</a></div><div class=\"lev1 toc-item\"><a href=\"#Smaller,-(or-equal),--die-from-larger\" data-toc-modified-id=\"Smaller,-(or-equal),--die-from-larger-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Smaller, (or equal),  die from larger</a></div><div class=\"lev2 toc-item\"><a href=\"#die4from6-Tests\" data-toc-modified-id=\"die4from6-Tests-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>die4from6 Tests</a></div><div class=\"lev1 toc-item\"><a href=\"#Larger-die-from-smaller\" data-toc-modified-id=\"Larger-die-from-smaller-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Larger die from smaller</a></div><div class=\"lev2 toc-item\"><a href=\"#Multiple-throws\" data-toc-modified-id=\"Multiple-throws-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Multiple throws</a></div><div class=\"lev2 toc-item\"><a href=\"#Creating-die13-from-die6\" data-toc-modified-id=\"Creating-die13-from-die6-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>Creating die13 from die6</a></div><div class=\"lev2 toc-item\"><a href=\"#Check-die13from6-created-from-die6-thrown-twice\" data-toc-modified-id=\"Check-die13from6-created-from-die6-thrown-twice-4.3\"><span class=\"toc-item-num\">4.3&nbsp;&nbsp;</span>Check die13from6 created from die6 thrown twice</a></div><div class=\"lev1 toc-item\"><a href=\"#Lets-generalise-to-n-from-m\" data-toc-modified-id=\"Lets-generalise-to-n-from-m-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Lets generalise to n from m</a></div><div class=\"lev2 toc-item\"><a href=\"#Check-die5_6-created-from-Die_n_from_m(5,-6)\" data-toc-modified-id=\"Check-die5_6-created-from-Die_n_from_m(5,-6)-5.1\"><span class=\"toc-item-num\">5.1&nbsp;&nbsp;</span>Check die5_6 created from Die_n_from_m(5, 6)</a></div><div class=\"lev2 toc-item\"><a href=\"#Check-die-created-from-Die_n_from_m(17,-6)\" data-toc-modified-id=\"Check-die-created-from-Die_n_from_m(17,-6)-5.2\"><span class=\"toc-item-num\">5.2&nbsp;&nbsp;</span>Check die created from Die_n_from_m(17, 6)</a></div><div class=\"lev2 toc-item\"><a href=\"#Check-die-created-from-Die_n_from_m(10,-3)\" data-toc-modified-id=\"Check-die-created-from-Die_n_from_m(10,-3)-5.3\"><span class=\"toc-item-num\">5.3&nbsp;&nbsp;</span>Check die created from Die_n_from_m(10, 3)</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Create an N-sided die from an M-sided die\n",
    "This is all about fair dice simulations. \n",
    "\n",
    "## What does fair mean? \n",
    "For the purposes of the simulation, a fair, X-sided die has equal probability of producing the numbers 1-to-N, inclusive, on a simulated roll.\n",
    "\n",
    "# A six-sided die simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on method randint in module random:\n",
      "\n",
      "randint(a, b) method of random.Random instance\n",
      "    Return random integer in range [a, b], including both end points.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "from random import randint \n",
    "\n",
    "help(randint)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def die6():\n",
    "    return randint(1, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How good is the simulation?\n",
    "I'll not go as far as Chi-squared tests, just show a histogram and summary data from rolling (calling) the die a few times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Series.describe: \n",
      "        Generates descriptive statistics that summarize the central tendency,\n",
      "        dispersion and shape of a dataset's distribution, excluding\n",
      "        ``NaN`` values.\n"
     ]
    }
   ],
   "source": [
    "from pandas import Series\n",
    "from collections import Counter\n",
    "from pprint import pprint as pp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "count = 500_000   # Sample count\n",
    "samples = Series(Counter(die6() for x in range(count)))\n",
    "\n",
    "print('Series.describe:', '\\n'.join(samples.describe.__doc__.split('\\n')[:4]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{1: 83239,\n",
      " 2: 83387,\n",
      " 3: 83948,\n",
      " 4: 83327,\n",
      " 5: 83092,\n",
      " 6: 83007}\n"
     ]
    },
    {
     "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": [
    "def show_me(values):\n",
    "    print(\"Map then graph rolled numbers to their frequencies:\")\n",
    "    pp(dict(values), width=20)\n",
    "    #print(\"\\nSummary stats:\")\n",
    "    #print(Series(list(Counter(dict(values)).elements())).describe())\n",
    "    values.sort_index(ascending=False).plot.barh(title=\"Frequencies of rolled numbers\")\n",
    "    plt.show()\n",
    "\n",
    "show_me(samples)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Well, that looks good enough."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Smaller, (or equal),  die from larger\n",
    "That's easy. Lets say we want to make a four sided die from the six sided one; all we do is select the first four possibilites from from the six sided die throws as valid outputs. If a five or a six is thrown by `die6` then throw again until it gives a `1` to `4` then return that number. We know that all the numbers from `die6` have the same probability of occuring **on _any_ throw**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def die4from6():\n",
    "    while True:\n",
    "        thrown = die6()\n",
    "        if 1 <= thrown <= 4:\n",
    "            return thrown"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## die4from6 Tests"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{1: 124766,\n",
      " 2: 124725,\n",
      " 3: 125224,\n",
      " 4: 125285}\n"
     ]
    },
    {
     "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": [
    "samples = Series(Counter(die4from6() for x in range(count)))\n",
    "show_me(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That looks good again."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Larger die from smaller\n",
    "\n",
    "That's a little more involved. Lets first take a concrete case of creating a thirteen-sided die from a six sided die.\n",
    "\n",
    "## Multiple throws\n",
    "If we throw the six sided die two times then there are `6*6 = 36` possible outcomes from the two throws. (We can keep the outcome order significant so a throw of `3` followed by a throw of `4` is distinct from a throw of `4` then a throw of `3`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{(1, 1): 13733,\n",
      " (1, 2): 13887,\n",
      " (1, 3): 13904,\n",
      " (1, 4): 13977,\n",
      " (1, 5): 13970,\n",
      " (1, 6): 14022,\n",
      " (2, 1): 13891,\n",
      " (2, 2): 13776,\n",
      " (2, 3): 14051,\n",
      " (2, 4): 13885,\n",
      " (2, 5): 13966,\n",
      " (2, 6): 13840,\n",
      " (3, 1): 13751,\n",
      " (3, 2): 14080,\n",
      " (3, 3): 13848,\n",
      " (3, 4): 13993,\n",
      " (3, 5): 13707,\n",
      " (3, 6): 13972,\n",
      " (4, 1): 13879,\n",
      " (4, 2): 13907,\n",
      " (4, 3): 13741,\n",
      " (4, 4): 13911,\n",
      " (4, 5): 13720,\n",
      " (4, 6): 13906,\n",
      " (5, 1): 13817,\n",
      " (5, 2): 14138,\n",
      " (5, 3): 13929,\n",
      " (5, 4): 14194,\n",
      " (5, 5): 13797,\n",
      " (5, 6): 14116,\n",
      " (6, 1): 13779,\n",
      " (6, 2): 13969,\n",
      " (6, 3): 13774,\n",
      " (6, 4): 13655,\n",
      " (6, 5): 13723,\n",
      " (6, 6): 13792}\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Number of distinct values from two throws is 36\n"
     ]
    }
   ],
   "source": [
    "samples = Series(Counter((die6(), die6()) \n",
    "                         for x in range(count)))\n",
    "show_me(samples)\n",
    "print(f'\\nNumber of distinct values from two throws is {len(samples)}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creating die13 from die6\n",
    "We can make die13 by \n",
    "\n",
    "* throwing die6 twice; \n",
    "* select and number the first 13 possibilites from the six sided die throws as valid outputs. \n",
    "  * If invalid possibilities are thrown then throw die6 __twice__ again until it gives a valid output, \n",
    "* then return the number associated with that valid combination. \n",
    "\n",
    "We know that all the two-throw number combinations from `die6` have the same probability of occuring so those selcted for die13 should also occur with the same probability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First thirteen of possible die6 2-throws:\n",
      "  ((1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1))\n"
     ]
    }
   ],
   "source": [
    "from itertools import product\n",
    "\n",
    "possible_2throws = tuple(product(range(1, 6+1), repeat=2))\n",
    "assert set(samples.keys()) == set(possible_2throws), 'Does not check with last sim result of two throws, above'\n",
    "\n",
    "print(f'First thirteen of possible die6 2-throws:\\n  {possible_2throws[:13]}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): 1,\n",
       " (1, 2): 2,\n",
       " (1, 3): 3,\n",
       " (1, 4): 4,\n",
       " (1, 5): 5,\n",
       " (1, 6): 6,\n",
       " (2, 1): 7,\n",
       " (2, 2): 8,\n",
       " (2, 3): 9,\n",
       " (2, 4): 10,\n",
       " (2, 5): 11,\n",
       " (2, 6): 12,\n",
       " (3, 1): 13}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "throws2output = {throw: n+1 for n, throw in enumerate(possible_2throws[:13])}\n",
    "throws2output"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def die13from6():\n",
    "    while True:\n",
    "        thrown = tuple(die6() for _ in range(2))\n",
    "        if thrown in throws2output:\n",
    "            return throws2output[thrown]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Check die13from6 created from die6 thrown twice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{1: 38260,\n",
      " 2: 38376,\n",
      " 3: 38480,\n",
      " 4: 38427,\n",
      " 5: 38356,\n",
      " 6: 38558,\n",
      " 7: 38590,\n",
      " 8: 38550,\n",
      " 9: 38869,\n",
      " 10: 38408,\n",
      " 11: 38473,\n",
      " 12: 38299,\n",
      " 13: 38354}\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXkAAAEICAYAAAC6fYRZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAAGL5JREFUeJzt3X2cXVV97/HP1+EhJERCSOBCgozhSR5UoLlWRSkSLBgi2AdfYq82VG9Tq22letVQfJWo9VVaq6X3tpVaoAFRFFGqF4tNKqZIK9AJBJIYkBAChERCxIRAWh7Cr3/slbAzzJyZ2XvNPsPO9/16ndfs5/U7a5/zmz1r71lLEYGZmbXTy7odgJmZjR4neTOzFnOSNzNrMSd5M7MWc5I3M2sxJ3kzsxZzkreukvSkpBkNlneQpJslbZX0+QzHO0/SLaX5kHRExWNV3jcnSUsk/e9ux2F57NHtAKwzSWuBg4DtpcVHRcT67kSUV0Ts23CR84BNwMvD/yRiuwFfyb80vD0i9i29XpTgJfkX9vAcBvx4uAleUs8ox9NaKjjHdJlPwEuUpN705/37JT0E3JSWv17Sv0vaLOkuSaeW9nmlpH9NTRWLJf21pKvTulMlretXxlpJp6fpl0maL+l+ST+TdK2kyf1imSvpIUmbJF1YOk6PpD9K+26VtFTSoWndziYKSXtL+ot0jEclXSppn7RuiqQb0vt6XNIPB0sgkt4o6T8kbUk/35iWLwTmAh9PzUSnD7DvQklflPRPkp4C3iJpP0lXSXpM0oOSPjmc5NXp/aT1H5O0QdJ6Se8b4lhLJH1G0r+lOlwkacowz90CSd+QdHXad7mkoyRdIGmjpIcl/XK/Ig+XdHuqw2/vONfpeJ0+Y0skfVbSvwHbgBkqmrTWpLIfkPS/hqo7yygi/BrDL2AtcPoAy3uBAK4CJgD7ANOAnwGzKX6BvzXNT037/Aj4ArA3cAqwFbg6rTsVWDdY2cD5wK3A9LT/3wHX9Ivl71McrwWeBo5J6z8GLAeOBpTWH5DWBXBEmr4E+A4wGZgI/H/gT9O6PwUuBfZMrzcDGqBeJgM/B95L0Rz57jS/o7yFwJ90qO+FwBbg5FSH41IdfzvF1Av8BHh/2v484JbS/sN9P2cCjwLHp/P31fK+A8S1BLgfOCrV8RLg4mGeuwXAfwFnpDq5CngAuDDV5W8DD/Qr65FSbN/khc/JUJ+xJcBDwHGprP2AJ4Cj0/qDgeO6/b3anV5dD8CvIU5Q8WV9EticXv+YlvempDCjtO0ngC/32/+fKa5eXwE8B0worfsqw0/yq4BZpXUHA8+mL/KOWKaX1t8OnJum7wXOGeT9BXAERfJ/Cji8tO4NO5IP8GmKRDtgEizt817g9n7LfgScl6YXMnSSv6o030PxC+vY0rLfAZak6fMYIMkP4/1cQUrSaf4ohk7ynyzNfxD43jDP3QJgcWnd29NnqifNT0xlTyqVVY7tWOCZVBeDfsZK+366tG4Cxef214B9uv192h1fbq55aXhHRExKr3f0W/dwafow4J3pz+jNkjYDb6JIyIcAP4+Ip0rbPziCGA4Dri8ddxXFzeCDStv8tDS9DdhxU/VQiqvQTqYC44GlpTK+l5YDfA5YDSxKf/rPH+Q4h/Di9/UgxRXocJXrdAqwV79jDud4Q72fQ/qVM5xzMVj9Dsejpen/BDZFxPbSPP2O1z+2PSnqotNn7EX7ps/bu4APABskfVfSq0YQt9XkJP/SV76B+DDFVdak0mtCRFwMbAD2lzShtP0rStNPUSQlYOcNx6ml9Q8Db+t37HER8cgwYnwYOHyIbTZRJJvjSsffL9LTNxGxNSI+GhEzKK5EPyJp1gDHWU+RiMpeQdH8MFzlOt1E8RdL+ZjDOV7H90NxPg7td8yqhjp3VfSP7VmK99TpM7bDLje1I+KfI+KtFL8I7qFo1rOGOMm3y9XA2yWdkW52jks35aZHxINAH/ApSXtJehNFstzhJ8A4SWdJ2hP4JEXb+w6XAp+VdBiApKmSzhlmXJcBn5F0pAqvkXRAeYOIeJ7iy/+Xkg5MZUyTdEaaniPpCEmiaOPdzq6Ple7wT8BRkn5D0h6S3kXR3HDDMGPdRbravZbivU9M7/8jFHXdab+O7ycd8zxJx0oaD1xUJb5kqHNXxXtKsX0auC7VxaCfsYEOouL/Es5OFxdPUzQTDXTebJQ4ybdIRDwMnAP8EfAYxVXXx3jhPP8G8IvA4xRJ5arSvlso2nkvo7hKfQooP7HxVxQ3ERdJ2kpxE/YXhxnaFyiS2iKKBH05xc3D/j5B0SRzq6QngH+huFkLcGSaf5Kijf1vI2LJAHXwM2AO8FGKG4IfB+ZExKZhxjqQ36eojzXALRT3Mq4Yxn6Dvp+IuJHixuxNaZubqgY3jHNXxZcp7k/8lOLm8x+ksob6jPX3MopzsZ7ic/dLKVZriCL8/yC7K0kLKG70vafbsZjZ6PCVvJlZiznJm5m1mJtrzMxazFfyZmYt1minVlOmTIne3t4mizQze8lbunTppoio9L8PjSb53t5e+vr6mizSzOwlT9JI/jt9F26uMTNrMSd5M7MWc5I3M2uxZkcTWn8nLNiv0SLNzBq1YEu3I9hFrSt5SVekkWVW5ArIzMzyqdtcs5BihBszMxuDaiX5iLiZomc5MzMbg0b9xqukeZL6JPU9ts1dKJiZNWnUk3xEfCkiZkbEzKnjNdrFmZlZiR+hNDNrMSd5M7MWq/WcvKRrgFOBKZLWARdFxOWD7nDIibDAfdeYmTWlVpKPiHfnCsTMzPJzc42ZWYs5yZuZtZiTvJlZiznJm5m1mJO8mVmLOcmbmbVYo/3JL39kC73zv9tkkWZmjVt78VndDmGnylfykg6V9ANJqyStlPThnIGZmVl9da7knwM+GhF3SJoILJW0OCJ+nCk2MzOrqfKVfERsiIg70vRWYBUwLVdgZmZWX5Ybr5J6gROB2wZYt7M/+e3bxtbYh2ZmbVc7yUvaF/gmcH5EPNF/fbk/+Z7xHsTbzKxJdQfy3pMiwX8lIr6VJyQzM8ulztM1Ai4HVkXEF/KFZGZmudS5kj8ZeC9wmqRl6TU7U1xmZpZB5UcoI+IWYESDtr562n70jaF/EjAzazt3a2Bm1mJO8mZmLeYkb2bWYk7yZmYt5iRvZtZiTvJmZi3mJG9m1mIeNMTMbBSMlYFD6nRrME7S7ZLuSoOGfCpnYGZmVl+dK/mngdMi4snUUdktkm6MiFszxWZmZjXV6dYggCfT7J7pFTmCMjOzPOp2NdwjaRmwEVgcER40xMxsDKmV5CNie0ScAEwHXifp+AG28aAhZmZdkuURyojYDCwBzsxxPDMzy6PO0zVTJU1K0/sApwP35ArMzMzqq/N0zcHAlZJ6KH5ZXBsRN3Tawf3Jm5k1q87TNXcDJ2aMxczMMnO3BmZmLeYkb2bWYk7yZmYt5iRvZtZiTvJmZi3mJG9m1mJO8mZmLdbooCF3bd3G//jBsiaLNDNr3E/fckK3Q9ip9pV86onyTkkd/9vVzMyal6O55sPAqgzHMTOzzOr2Jz8dOAu4LE84ZmaWU90r+UuAjwPPD7ZBedCQ57dsrlmcmZmNRJ2uhucAGyNiaaftyoOGvGy/SVWLMzOzCupcyZ8MnC1pLfA14DRJV2eJyszMsqic5CPigoiYHhG9wLnATRHxnmyRmZlZbY0+J//aiePpG0PPj5qZtV2WJB8RSyjGeDUzszHE3RqYmbWYk7yZWYs5yZuZtZiTvJlZiznJm5m1mJO8mVmLNfqc/Naty/n+TYc3WaSZWeNmnXZ/t0PYqVaST10abAW2A89FxMwcQZmZWR45ruTfEhGbMhzHzMwyc5u8mVmL1U3yASyStFTSvBwBmZlZPnWba06OiPWSDgQWS7onIm4ub5CS/zyAAw9s9D6vmdlur9aVfESsTz83AtcDrxtgm52Dhkya5NYhM7Mm1RkZaoKkiTumgV8GVuQKzMzM6qvTfnIQcL2kHcf5akR8L0tUZmaWReUkHxFrgNeOZJ+JE1/NrNP6qhZpZmYj5EZyM7MWc5I3M2sxJ3kzsxZzkjczazEneTOzFnOSNzNrMSd5M7MWa7QzmfXr17NgwYImizQz64qxkutqXclLmiTpOkn3SFol6Q25AjMzs/rqXsn/FfC9iPh1SXsB4zPEZGZmmVRO8pJeDpwCnAcQEc8Az+QJy8zMcqjTXDMDeAz4B0l3Sros9Ua5C0nzJPVJ6tu2bVuN4szMbKTqJPk9gJOAL0bEicBTwPz+G5X7kx8/3q05ZmZNqpPk1wHrIuK2NH8dRdI3M7MxonKSj4ifAg9LOjotmgX8OEtUZmaWRd2na34f+Ep6smYN8Fv1QzIzs1wUEY0VNnPmzOjr86AhZmYjIWlpRMyssq+7NTAzazEneTOzFnOSNzNrMSd5M7MWc5I3M2sxJ3kzsxZzkjcza7FGBw155pEnWTf/h00WaWbWNdMvfnO3Q6h+JS/paEnLSq8nJJ2fMzgzM6un8pV8RNwLnAAgqQd4BLg+U1xmZpZBrjb5WcD9EfFgpuOZmVkGuZL8ucA1A60oDxry+LbNmYozM7PhqJ3kUw+UZwPfGGh9edCQyeMn1S3OzMxGIMeV/NuAOyLi0QzHMjOzjHIk+XczSFONmZl1V63n5CWNB94K/M5wtt9r2r5j4rlRM7PdRa0kHxHbgAMyxWJmZpm5WwMzsxZzkjczazEneTOzFnOSNzNrMSd5M7MWc5I3M2sxJ3kzsxZrdNCQR9es5vPvmtNkkWZmXfHRr9/Q7RCAmlfykv5Q0kpJKyRdI2lcrsDMzKy+OiNDTQP+AJgZEccDPRRdDpuZ2RhRt01+D2AfSXsA44H19UMyM7NcKif5iHgE+AvgIWADsCUiFvXfrjxoyFNPP1M9UjMzG7E6zTX7A+cArwQOASZIek//7cqDhkzYe6/qkZqZ2YjVaa45HXggIh6LiGeBbwFvzBOWmZnlUCfJPwS8XtJ4SaIYzHtVnrDMzCyHys/JR8Rtkq4D7gCeA+4EvtRpn4NmHDFmnh01M9sd1B005CLgokyxmJlZZu7WwMysxZzkzcxazEnezKzFnOTNzFrMSd7MrMWc5M3MWqzR/uQ3PriVv/nATU0WaWbWdR+69LSulV23P/kPp77kV0o6P1dQZmaWR50Oyo4Hfht4HfBaYI6kI3MFZmZm9dW5kj8GuDUitkXEc8C/Ar+SJywzM8uhTpJfAZwi6QBJ44HZwKF5wjIzsxzqdFC2StKfAYuBJ4G7KDoq24WkecA8gP33PbBqcWZmVkGtG68RcXlEnBQRpwCPA/cNsM3OQUP2HTepTnFmZjZCtR6hlHRgRGyU9ArgV4E35AnLzMxyqPuc/DclHQA8C3woIn6eISYzM8ukbn/ybx7J9gceNrGr/xRgZra7cbcGZmYt5iRvZtZiTvJmZi3mJG9m1mJO8mZmLeYkb2bWYk7yZmYt1uigIf+1YiWrXnVMk0WamTXumHtWdTuEnYa8kpd0haSNklaUlk2WtFjSfenn/qMbppmZVTGc5pqFwJn9ls0Hvh8RRwLfT/NmZjbGDJnkI+Jmih4my84BrkzTVwLvyByXmZllUPXG60ERsQEg/Ry0o3hJ8yT1Sep7fPuLups3M7NRNOpP15T7k5/c0+h9XjOz3V7VJP+opIMB0s+N+UIyM7Ncqib57wBz0/Rc4Nt5wjEzs5yGbD+RdA1wKjBF0jrgIuBi4FpJ7wceAt45nMLGHX8cx/T1VY/WzMxGZMgkHxHvHmTVrMyxmJlZZu7WwMysxZzkzcxazEnezKzFnOTNzFrMSd7MrMWc5M3MWsxJ3sysxRrtTGblz1by6itf3WSRZmZdsXzu8m6HAFQfNOSdklZKel7SzNEN0czMqqo6aMgK4FeBm3MHZGZm+QynW4ObJfX2W7YKQNLoRGVmZlmM+o3X8qAh27duH+3izMyspNFBQ3om9ox2cWZmVuJHKM3MWsxJ3sysxaoOGvI48P+AqcB3JS2LiDOGOtZxBxxH31wPGmJm1pQ6g4ZcnzkWMzPLzM01ZmYt5iRvZtZiTvJmZi3mJG9m1mJO8mZmLeYkb2bWYk7yZmYt1uigIay/Exbs12iRZmaNWrCl2xHsouqgIZ+TdI+kuyVdL2nS6IZpZmZVVB00ZDFwfES8BvgJcEHmuMzMLIMhk3xE3EzRV0152aKIeC7N3gpMH4XYzMysphw3Xt8H3DjYyvKgIY9tiwzFmZnZcNVK8pIuBJ4DvjLYNuVBQ6aO93CBZmZNqvx0jaS5wBxgVkT4Et3MbAyqlOQlnQl8AviliNiWNyQzM8ul6qAhFwB7A4slAdwaER8YsrRDToQFHjTEzKwpVQcNuXwUYjEzs8zcrYGZWYs5yZuZtZiTvJlZiznJm5m1mJO8mVmLOcmbmbVYo/3JL39kC73zv9tkkWZmjVt78VndDmGnqv3Jfyb1Jb9M0iJJh4xumGZmVkXV/uQ/FxGviYgTgBuAP84dmJmZ1Ve1P/knSrMTAHdQZmY2BtXphfKzwG8CW4C3dNhuHjAPoOflU6sWZ2ZmFVR+uiYiLoyIQyn6kv+9Dtvt7E++Z7wH8TYza1KORyi/CvxahuOYmVlmlZK8pCNLs2cD9+QJx8zMcqran/xsSUcDzwMPAkP3JW9mZo1TkyP3zZw5M/r6PGiImdlISFoaETOr7OtuDczMWsxJ3sysxZzkzcxarNE2eUlbgXsbK7CaKcCmbgcxBMeYh2PMwzHm0SnGwyKi0n+TNtoLJXBv1ZsHTZHU5xjrc4x5OMY8ducY3VxjZtZiTvJmZi3WdJL/UsPlVeEY83CMeTjGPHbbGBu98WpmZs1yc42ZWYs5yZuZtVgjSV7SmZLulbRa0vwmyuxX/lpJy9OYtH1p2WRJiyXdl37un5ZL0v9Nsd4t6aTSceam7e+TNLdmTAONnZstJkm/kN7z6rSvMsW4QNIjqS6XSZpdWndBKu9eSWeUlg94/iW9UtJtKfavS9qrQoyHSvqBpFWSVkr6cFo+ZuqyQ4xjpi4ljZN0u6S7Uoyf6nRcSXun+dVpfW/V2DPEuFDSA6V6PCEt78r3Jh2nR9Kdkm5I892rx4gY1RfQA9wPzAD2Au4Cjh3tcvvFsBaY0m/ZnwPz0/R84M/S9GzgRkDA64Hb0vLJwJr0c/80vX+NmE4BTgJWjEZMwO3AG9I+NwJvyxTjAuD/DLDtsenc7g28Mp3znk7nH7gWODdNXwr8boUYDwZOStMTgZ+kWMZMXXaIcczUZXpv+6bpPYHbUv0MeFzgg8Clafpc4OtVY88Q40Lg1wfYvivfm3Scj1CMtXFDp/PTRD02cSX/OmB1RKyJiGeArwHnNFDuUM4BrkzTVwLvKC2/Kgq3ApMkHQycASyOiMcj4ufAYl48wPmwxQBj5+aKKa17eUT8KIpPzFWlY9WNcTDnAF+LiKcj4gFgNcW5H/D8pyuk04DrBni/I4lxQ0Tckaa3AquAaYyhuuwQ42Aar8tUH0+m2T3TKzoct1y/1wGzUhwjij1TjIPpyvdG0nTgLOCyNN/p/Ix6PTaR5KcBD5fm19H5Az4aAlgkaamKMWcBDoqIDVB8CYED0/LB4m3ifeSKaVqaHq1Yfy/9+XuFUjNIhRgPADZHxHO5Ykx/6p5IcYU3JuuyX4wwhuoyNTEsAzZSJL77Oxx3Zyxp/ZYUx6h+f/rHGBE76vGzqR7/UtLe/WMcZiy5zvUlwMcpxtuAzudn1OuxiSQ/UJtW089tnhwRJwFvAz4k6ZQO2w4Wbzffx0hjGs1YvwgcDpwAbAA+n5Z3NUZJ+wLfBM6PiCc6bTrCeLLFOUCMY6ouI2J7RJwATKe4Yjymw3HHRIySjgcuAF4F/E+KJphPdCtGSXOAjRGxtLy4w3FHPcYmkvw64NDS/HRgfQPl7hQR69PPjcD1FB/gR9OfZ6SfG9Pmg8XbxPvIFdO6NJ091oh4NH3Rngf+nqIuq8S4ieLP5z36LR8xSXtSJM+vRMS30uIxVZcDxTgW6zLFtRlYQtGOPdhxd8aS1u9H0bTXyPenFOOZqTksIuJp4B+oXo85zvXJwNmS1lI0pZxGcWXfvXrs1GCf40XRCdoaipsHO24UHDfa5ZbKnwBMLE3/O0Vb+ufY9cbcn6fps9j1Zs3t8cLNmgcobtTsn6Yn14ytl11vamaLCfiPtO2OG0izM8V4cGn6DynaDQGOY9cbRWsobhINev6Bb7DrzagPVohPFG2nl/RbPmbqskOMY6YuganApDS9D/BDYM5gxwU+xK43DK+tGnuGGA8u1fMlwMXd/t6kY53KCzdeu1aPTSXa2RRPFNwPXNhEmaWyZ6SKuAtYuaN8inav7wP3pZ87TrKAv0mxLgdmlo71PoobIKuB36oZ1zUUf6I/S/Hb+f05YwJmAivSPn9N+u/mDDF+OcVwN/Addk1UF6by7qX0VMJg5z+dm9tT7N8A9q4Q45so/ly9G1iWXrPHUl12iHHM1CXwGuDOFMsK4I87HRcYl+ZXp/UzqsaeIcabUj2uAK7mhSdwuvK9KR3rVF5I8l2rR3drYGbWYv6PVzOzFnOSNzNrMSd5M7MWc5I3M2sxJ3kzsxZzkjczazEneTOzFvtvB+PzAO3IKywAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "samples = Series(Counter(die13from6() for x in range(count)))\n",
    "show_me(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lets generalise to n from m\n",
    "Tidy global variables into a class and make its instances callable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Die_n_from_m():\n",
    "    def __init__(self, n, m):\n",
    "        self.n = n\n",
    "        self.m = m\n",
    "        # Work out ho many die_m rolls will be needed\n",
    "        rollcount, possibilities = 1, m\n",
    "        while possibilities < n:\n",
    "            rollcount, possibilities = rollcount + 1, possibilities * m\n",
    "        self.rollcount = rollcount\n",
    "        #\n",
    "        possible_throws = tuple(product(range(1, m + 1), repeat=rollcount))[:n]\n",
    "        self.throws2output = {throw: n+1 for n, throw in enumerate(possible_throws)}\n",
    "\n",
    "    def die_m(self):\n",
    "        return randint(1, self.m)    \n",
    "\n",
    "    def die_n(self):\n",
    "        while True:\n",
    "            thrown = tuple(self.die_m() for _ in range(self.rollcount))\n",
    "            if thrown in self.throws2output:\n",
    "                return self.throws2output[thrown] \n",
    "\n",
    "    __call__ = die_n\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "d5_6 = Die_n_from_m(5, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d5_6()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Check die5_6 created from Die_n_from_m(5, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{1: 100063,\n",
      " 2: 99908,\n",
      " 3: 100384,\n",
      " 4: 99886,\n",
      " 5: 99759}\n"
     ]
    },
    {
     "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": [
    "samples = Series(Counter(d5_6() for x in range(count)))\n",
    "show_me(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Check die created from Die_n_from_m(17, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{1: 29751,\n",
      " 2: 29224,\n",
      " 3: 29494,\n",
      " 4: 29372,\n",
      " 5: 29534,\n",
      " 6: 29077,\n",
      " 7: 29438,\n",
      " 8: 29573,\n",
      " 9: 29817,\n",
      " 10: 29182,\n",
      " 11: 29079,\n",
      " 12: 29364,\n",
      " 13: 29690,\n",
      " 14: 29256,\n",
      " 15: 29399,\n",
      " 16: 29558,\n",
      " 17: 29192}\n"
     ]
    },
    {
     "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": [
    "die = Die_n_from_m(17, 6)\n",
    "samples = Series(Counter(die() for x in range(count)))\n",
    "show_me(samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Check die created from Die_n_from_m(10, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Map then graph rolled numbers to their frequencies:\n",
      "{1: 50081,\n",
      " 2: 49733,\n",
      " 3: 50368,\n",
      " 4: 50142,\n",
      " 5: 49675,\n",
      " 6: 50060,\n",
      " 7: 49977,\n",
      " 8: 50023,\n",
      " 9: 49943,\n",
      " 10: 49998}\n"
     ]
    },
    {
     "data": {
      "image/png": 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AGcAnxlMdSdKwRurUrqptwCHznX+/x53ECdPToxQpSdoJ7xSVpJ4w0CWpJwx0SeoJA12SesJAl6SeMNAlqScMdEnqCQNdknrCQJeknjDQJaknJvr/bK/97rU8/v2Pn2SRkrQorjn7momX6RG6JPXEqP9t8b8kuTbJhiQXJ9lvXBWTJC3M0IGe5Gjgt4CpqnocsIzuUXSSpEUwapfLXsD+SfYClgObRq+SJGkYQwd6Vd0B/DHwLeBO4N6qunTmfIMPiX5w64PD11SSNKdRulwOBl4MPBI4CjggyStmzjf4kOhlK5YNX1NJ0pxG6XJ5DvDNqvpOVf2Y7jF0Tx9PtSRJCzVKoH8LeFqS5UkCnA5sHE+1JEkLNUof+hXAx4CvAte0da0dU70kSQs06kOi3wa8bUx1kSSNYKK3/p90yElMnz09ySIlacnw1n9J6gkDXZJ6wkCXpJ4w0CWpJwx0SeoJA12SesJAl6SeMNAlqScMdEnqCQNdknpiorf+s+lrcN5BEy1SkhbFefdOvEiP0CWpJ0YK9CSvTbIhybVJXjeuSkmSFm6UR9A9Dvh14CnAE4CzkjxmXBWTJC3MKEfoJwBfqaptVfUA8PfAvx5PtSRJCzVKoG8ATk1ySJLlwJnAsTNnSrImyXSS6e9sqxGKkyTNZeirXKpqY5I/AC4D7gOuBh6YZb61tEfTTR21zESXpN1kpJOiVfXeqnpyVZ0K3APcMJ5qSZIWaqTr0JMcXlWbkzwc+DfAyeOpliRpoUa9sejjSQ4Bfgy8pqq+N4Y6SZKGMFKgV9Uzx1URSdJoJnvr/1FPgvOmJ1qkJC0V3vovST1hoEtSTxjoktQTBrok9YSBLkk9YaBLUk8Y6JLUEwa6JPWEgS5JPTHRO0WvueNeVp/7t5MsUpIWxS3vfMHEy/QIXZJ6wkCXpJ7YZaAneV+SzUk2DIxbleSyJDe0nwfv3mpKknZlPkfoFwLPmzHuXOBzVfUY4HPtvSRpEe0y0KvqC3SPlxv0YuD9bfj9wEvGXC9J0gIN24d+RFXdCdB+Hr6zGZOsSTKdZPrBbfcOWZwkaVd2+0nRqlpbVVNVNbVs+UG7uzhJWrKGDfS7khwJ0H5uHl+VJEnDGDbQ/wY4uw2fDXxqPNWRJA1rPpctXgx8GTg+ye1JXgW8EzgjyQ3AGe29JGkRpaomVtjU1FRNT/uQaElaiCTrq2pqV/N5p6gk9YSBLkk9YaBLUk8Y6JLUEwa6JPXERK9ySbIVuH5iBe5ZDgXuXuxKLJKl3HZY2u237ePxiKo6bFczTfSJRcD187n0po+STNv2pWkpt9+2T7btdrlIUk8Y6JLUE5MO9LUTLm9PYtuXrqXcfts+QRM9KSpJ2n3scpGknjDQJaknJhLoSZ6X5PokNyZ5SD9QOsn7kmxOsmFg3KoklyW5of08uI1Pkj9r7f7nJE8eWObsNv8NSc4eGP8vk1zTlvmzJJlsC2eX5Ngkn0+yMcm1SV7bxve+7QBJ9ktyZZKrW/vf3sY/MskVrS0fTrJPG79ve39jm756YF1vbuOvT/LcgfF79PckybIkX0tySXu/JNqe5Ja2X16VZLqN2zP3+6rarS9gGXATcBywD3A1cOLuLnc3tudU4MnAhoFxfwic24bPBf6gDZ8JfBoI8DTgijZ+FXBz+3lwGz64TbsSOLkt82ng+Yvd5lavI4Ent+EVwDeAE5dC21vdAhzYhvcGrmjt+gjwsjb+AuA/t+FXAxe04ZcBH27DJ7bvwL7AI9t3Y9lD4XsCvB74EHBJe78k2g7cAhw6Y9weud9PYmOcDHx24P2bgTcv9oc0YptWs2OgXw8c2YaPpLuBCuDdwMtnzge8HHj3wPh3t3FHAtcNjN9hvj3pRfeUqjOWaNuXA18Fnkp3J+BebfxP93Xgs8DJbXivNl9m7v/b59vTvyfAMcDngNOAS1pblkrbb+HnA32P3O8n0eVyNHDbwPvb27g+OaKq7gRoPw9v43fW9rnG3z7L+D1K+xP6SXRHqUum7a3L4Sq6Z+heRndUuaWqHmizDNb5p+1s0+8FDmHh22VPcT7wJuAn7f0hLJ22F3BpkvVJ1rRxe+R+P4lb/2frD1oq10rurO0LHb/HSHIg8HHgdVX1/Tm6+3rX9qp6EHhikpXAJ4ETZput/VxoO2c7uNoj2p/kLGBzVa1P8qzto2eZtXdtb06pqk1JDgcuS3LdHPMu6n4/iSP024FjB94fA2yaQLmTdFeSIwHaz81t/M7aPtf4Y2YZv0dIsjddmH+wqj7RRi+Jtg+qqi3AOro+0pVJth8YDdb5p+1s0w8C7mHh22VPcArwoiS3AH9N1+1yPkuj7VTVpvZzM90v8qewp+73E+h/2ovuBMAj+dkJj5MWu19sxDatZsc+9D9ixxMkf9iGX8COJ0iubONXAd+kOzlycBte1ab9U5t3+wmSMxe7va1eAS4Czp8xvvdtb3U7DFjZhvcHvgicBXyUHU8MvroNv4YdTwx+pA2fxI4nBm+mOyn4kPieAM/iZydFe9924ABgxcDwPwLP21P3+0ltlDPproq4CXjLYn9II7blYuBO4Md0v11fRdc/+DnghvZz+wcV4C9au68BpgbW82vAje31qwPjp4ANbZk/p93Nu9gv4Bl0fwr+M3BVe525FNre6vZLwNda+zcA/7WNP47uKoUbW8Dt28bv197f2KYfN7Cut7Q2Xs/AFQ0Phe8JOwZ679ve2nh1e127vW576n7vrf+S1BPeKSpJPWGgS1JPGOiS1BMGuiT1hIEuST1hoEtSTxjoktQT/x9VW3K/7qsD/AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "die = Die_n_from_m(10, 3)\n",
    "samples = Series(Counter(die() for x in range(count)))\n",
    "show_me(samples)"
   ]
  },
  {
   "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": "3.6.8"
  },
  "nav_menu": {},
  "toc": {
   "navigate_menu": true,
   "number_sections": true,
   "sideBar": true,
   "threshold": 6,
   "toc_cell": true,
   "toc_section_display": "block",
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}