{
 "metadata": {
  "name": "1.Rolling a die.ipynb"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Uniform Random Probabilities\n",
      "## *If I roll a die 100 times, how many times will it come up a 6?*\n",
      "---"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import random\n",
      "import math\n",
      "\n",
      "num_rand_vals = 100\n",
      "min_value = 1\n",
      "max_value = 6\n",
      "\n",
      "rand_vals = []\n",
      "\n",
      "print \"Roll: number\"\n",
      "for i in xrange(num_rand_vals):\n",
      "    val = random.randint(min_value, max_value)\n",
      "    rand_vals.append(val)\n",
      "    if (i < 10):\n",
      "        print \"%d: %d\" % (i, val)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Roll: number\n",
        "0: 4\n",
        "1: 3\n",
        "2: 2\n",
        "3: 3\n",
        "4: 3\n",
        "5: 6\n",
        "6: 4\n",
        "7: 2\n",
        "8: 1\n",
        "9: 4\n"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "min_seen = min(rand_vals)\n",
      "max_seen = max(rand_vals)\n",
      "mean_seen = (sum(rand_vals)+0.0) / len(rand_vals)\n",
      "\n",
      "print \"When simulating %d random values between %d and %d\" % (num_rand_vals, min_value, max_value)\n",
      "print \"The smallest I saw was %d\" % (min_seen)\n",
      "print \"The biggest I saw was %d\" % (max_seen)\n",
      "print \"And the mean value was %.02f\" % (mean_seen)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "When simulating 100 random values between 1 and 6\n",
        "The smallest I saw was 1\n",
        "The biggest I saw was 6\n",
        "And the mean value was 3.41\n"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Plot the individual vals\n",
      "plt.figure()\n",
      "plt.xlabel(\"Roll Index\")\n",
      "plt.ylabel(\"Result\")\n",
      "plt.scatter(range(num_rand_vals), rand_vals)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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8eBCPPPJITdQmhBDCAyr8qZ7CwkLXZZ6VK1di5MiR1V9cPtUjhBCV5rFP9RQU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\nU8dh1KiRVV77p59+wl//+g9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QwstI8Is6TaPRIDY2Fh06dMDjjz+OgoKCMsffe++9+PTT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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10b6f98d0>"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## summarize how many times we saw each value\n",
      "tally = [0 for x in xrange(max_value+1)]\n",
      "for i in rand_vals:\n",
      "    tally[i] += 1\n",
      "\n",
      "print \"value #times_seen\"\n",
      "for t in xrange(len(tally)):\n",
      "    print \"%d: %d\" % (t, tally[t])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "value #times_seen\n",
        "0: 0\n",
        "1: 14\n",
        "2: 22\n",
        "3: 15\n",
        "4: 19\n",
        "5: 18\n",
        "6: 12\n"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Why does 0 have 0?\n",
      "---"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"We expected to see the number 6 %.02f percent of the time = %d times\" % (100./6, num_rand_vals/6)\n",
      "print \"We actually saw 6 %d times\" % (tally[6])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "We expected to see the number 6 16.67 percent of the time = 16 times\n",
        "We actually saw 6 12 times\n"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## summarize the tally\n",
      "mean_tally = (sum(tally) + 0.) / (len(tally)-1) ## skip 0\n",
      "min_tally = tally[1]\n",
      "\n",
      "sumdiff = 0.0\n",
      "for i in xrange(min_value, max_value+1):\n",
      "    diff = (tally[i] - mean_tally) ** 2\n",
      "    sumdiff += diff\n",
      "    if (tally[i] < min_tally):\n",
      "        min_tally = tally[i]\n",
      "\n",
      "sumdiff /= (len(tally)-1)\n",
      "stdev_tally = math.sqrt(sumdiff)\n",
      "\n",
      "print \"the minimum tally was %d\" % (min_tally)\n",
      "print \"the maximum tally was %d\" % (max(tally))\n",
      "print \"the mean tally was %.02f +/- %.2f\" %  (mean_tally, stdev_tally)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "the minimum tally was 12\n",
        "the maximum tally was 22\n",
        "the mean tally was 16.67 +/- 3.35\n"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## plot the tally, a histogram\n",
      "plt.figure()\n",
      "plt.xlabel(\"Value\")\n",
      "plt.ylabel(\"Number of occurences\")\n",
      "plt.bar(range(max_value+1), tally)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10b820290>"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## The hist function will do everything for us in one step\n",
      "plt.figure()\n",
      "plt.xlabel(\"Value\")\n",
      "plt.ylabel(\"Number of occurences\")\n",
      "plt.hist(rand_vals, bins=range(max_value+2))\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10b822490>"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## The hist function can also plot the probability of each number, i.e. the density function\n",
      "plt.figure()\n",
      "plt.xlabel(\"Value\")\n",
      "plt.ylabel(\"Percent of occurences\")\n",
      "plt.hist(rand_vals, bins=range(max_value+2), normed=True)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10b6f9cd0>"
       ]
      }
     ],
     "prompt_number": 49
    }
   ],
   "metadata": {}
  }
 ]
}