{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Birthday problem simulated\n",
    "\n",
    "Let's say we can't figure out how to formally calculate the probability that 2 people out of N have the same birthday (ignoring leap years). Not to worry: we can simulate it!\n",
    "\n",
    "First let's write a function to simulate the number of people who have the same birthday out of N:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random\n",
    "from collections import defaultdict\n",
    "\n",
    "def num_people_same_birthday(n):\n",
    "    bdays = defaultdict(int)\n",
    "    for i in range(n):\n",
    "        bdays[random.randrange(0, 365)] += 1\n",
    "    return len([k for (k, v) in bdays.items() if v > 1])    \n",
    "\n",
    "num_people_same_birthday(60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That doesn't give as an emperical probability though, we need to run some trials to do that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.992"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def prob_n_people_have_same_birthday(n, *, num_trials):\n",
    "    num_positive_trials = len([1 for i in range(num_trials) if num_people_same_birthday(n) > 0])\n",
    "    return float(num_positive_trials) / num_trials\n",
    "\n",
    "prob_n_people_have_same_birthday(60, num_trials=1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that because the liklihood is high, we need 1000 trials to suss out that it's not 100%; here's what we get if we use 10 a couple times:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_n_people_have_same_birthday(60, num_trials=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_n_people_have_same_birthday(60, num_trials=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just to round out the \"fun\", let's see a bar chart of % of time we expect people to have the same birthday with groups ranging from 5 to 50 (counting by 5)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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L4bOUOqIYZJx0u1b1lHmfIe9Yf0kpX/dU6jDics1/cmHc9C3pSKeenTsTqTOp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\nAAAACkMwDwAAAABAYQjmAQA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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10659bfd0>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 280,
       "width": 505
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'retina'\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "group_sizes = list(range(5, 55, 5))\n",
    "probs = [prob_n_people_have_same_birthday(n, num_trials=1000) for n in group_sizes]\n",
    "plt.xlabel(\"Group Size\")\n",
    "plt.ylabel(\"Prob\")\n",
    "plt.title(\"Probability that among given group size at least 2 people have the same birthday\")\n",
    "\n",
    "plt.bar(group_sizes, probs, width=2.5)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now let's see a histogram of the number of people with the same birthday among 60 people out of 1000 trials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2q+f2OVV1wpp1Tkry1CQ3Jnnt0agXAAD2oh3dRrKqHpPkMfPb1ctcTq+q35//+6vd/Zvz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\nYCACPAAADOT/B8/dq6XtyH0SAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10659b5f8>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 255,
       "width": 376
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist([num_people_same_birthday(60) for i in range(1000)])\n",
    "None"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The histogram shows that the probability of there being zero people is unlikely, hence the overall probability of there being at least 1 is high."
   ]
  }
 ],
 "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": "3.4.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}