{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`numpy` has useful functions for generating random numbers and maninulating arrays.\n",
      "`matplotlib` has useful functions for plotting things."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will explicitly model the process of picking a bridge and then picking a creature, rejecting any trial in which a gnome is picked.\n",
      "\n",
      "`numpy.random.choice` is a convenient way to pick from a set of objects with specified weights (i.e draw from a [categorical distribution](http://en.wikipedia.org/wiki/Categorical_distribution)). "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "bridges = ['TGGGG', 'TTGGG', 'GGGGG']\n",
      "ps = [0.2, 0.2, 0.6]\n",
      "trials = []\n",
      "\n",
      "while len(trials) < 100000:\n",
      "    bridge = np.random.choice(bridges, p=ps)\n",
      "    creature = np.random.choice(list(bridge))\n",
      "    if creature == 'T':\n",
      "        if bridge.count('T') == 1:\n",
      "            trials.append(1)\n",
      "        else:\n",
      "            trials.append(0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`trials` is now a list of 0's and 1's that we need to convert into a list of \"fraction of safe passages so far\"."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cumulative_fractions = np.cumsum(trials) / (np.arange(len(trials)) + 1.)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.semilogx(cumulative_fractions)\n",
      "plt.ylim(0, 1)\n",
      "plt.axhline(1. / 3, color='red', linestyle='--')\n",
      "plt.xlabel('Number of trials')\n",
      "plt.ylabel('Cumulative fraction of safe passages');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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++ANHjx7F9u3bcfr06RJfZD6NGskJ/HSNRRw/btiTQ9WqyQHfNWtMCg9CyLUc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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9a9d90c>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The simulation takes a while, in part because we need to generate about $1 / (0.2 * 0.2 + 0.4 * 0.2) \\approx 8.3$ trials per troll captured (can you see why?) and in part because `np.random.choice` is just slow. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit np.random.choice(bridges, p=ps)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10000 loops, best of 3: 143 \u00b5s per loop\n"
       ]
      }
     ],
     "prompt_number": 6
    }
   ],
   "metadata": {}
  }
 ]
}