dice_combinations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3bf75b53-c988-409b-8d76-93acd761b895",
   "metadata": {},
   "source": [
    "# Probabilities of dice totals\n",
    "    by Gerrit Nowald\n",
    "In the boardgame [The Settlers of Catan](https://simple.wikipedia.org/wiki/The_Settlers_of_Catan), the ressources are distributed based on the total of two 6-sided dice. The total 7 moves the bandit, which is used to steal ressources from other players. The other totals, ranging from 2 to 12, allow players to gain ressources.\n",
    "\n",
    "To get the probability of a dice total, the number it occurs is divided by the number of all possible combinations. E.g. for two 6-sided dice, there are a total of 6*6=36 possible combinations.\n",
    "\n",
    "This Notebook shows a Python implementation to calculate and plot the probability distribution for all dice totals, which is also applicable to arbitrary many, arbitrary-sided dice.\n",
    "\n",
    "See also this blog post: https://gerritnowald.wordpress.com/2022/04/20/probabilities-of-dice-totals/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d09940d2-8f5e-4aa2-8c76-6da2cca53f54",
   "metadata": {},
   "outputs": [],
   "source": [
    "import itertools\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from matplotlib.ticker import MaxNLocator"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f38ab23-ede9-4260-a92d-a359f6a8b6d5",
   "metadata": {},
   "source": [
    "## input: sides of dice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6a927ed4-74cb-459e-b85c-7d0befaceb52",
   "metadata": {},
   "outputs": [],
   "source": [
    "dice = [6, 6]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0bf7779b-027c-4b59-bc99-346225ddbe08",
   "metadata": {},
   "source": [
    "## calculation\n",
    "- list of lists of all possible eyes for each die:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "78e72bc7-bde9-4794-8699-afe8b730c495",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6]]\n"
     ]
    }
   ],
   "source": [
    "eyes = [ [eye+1 for eye in range(die)] for die in dice ]\n",
    "print(eyes)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9e8c98c-e0bd-4e7a-bb33-b3236724209c",
   "metadata": {},
   "source": [
    "- all possible result combinations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0a11f10a-1ade-45a8-ac1a-839fe2819706",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(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), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)]\n"
     ]
    }
   ],
   "source": [
    "combinations = list(itertools.product(*eyes))\n",
    "print(combinations)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52292d1b-aaf6-425e-929d-02bfbae38740",
   "metadata": {},
   "source": [
    "- dice total for each combination:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "316217e4-ecc7-47ce-a4e1-95dfe37ae50e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12]\n"
     ]
    }
   ],
   "source": [
    "dicetotal = [sum(combination) for combination in combinations]\n",
    "print(dicetotal)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c765993-b4ee-4886-b63e-93eb60a5aa5b",
   "metadata": {},
   "source": [
    "- probability distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "480bca06-b0a2-43b6-bfb3-93297c5c78bf",
   "metadata": {},
   "outputs": [],
   "source": [
    "N = np.histogram(dicetotal, bins = np.arange(min(dicetotal), min(dicetotal) + len(set(dicetotal)) + 1), density=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "616ecc5d-05d0-4038-be44-90f9e6915db8",
   "metadata": {},
   "source": [
    "## plot probability distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "20650e6a-9084-402c-9592-6b5a0ac51cf1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = plt.figure().gca()\n",
    "plt.bar(N[1][:-1], N[0]*100, width=0.8)\n",
    "plt.title(f'probability distribution for dice: {dice}')\n",
    "plt.xlabel('dice totals')\n",
    "plt.ylabel('probability / %')\n",
    "ax.xaxis.set_major_locator(MaxNLocator(integer=True))   # integer ticks"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "729b5f5c-bd63-4486-9b7a-fe67ede2ff34",
   "metadata": {},
   "source": [
    "The probability is highest for the total 7 and decreases towards the extreme values. The probability distribution is symmetrical, e.g. getting a total of 4 is equally likely as getting a total of 10."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.10.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}