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   "source": [
    "## Problem setup:\n",
    "\n",
    "1. 13,066,047 voters arrive to the polls.\n",
    "2. $p_{yes}\\%$ of them intend to vote \"Yes\", $(1-p_{yes})\\%$ of them intend to vote \"No.\"\n",
    "3. Each voter casts an invalid (unmarked or void) ballot with probability $p_{invalid}\\%$.\n",
    "4. Of the valid ballots, the poll workers misclassify the vote with probability $p_{misclassification}\\%$.\n",
    "5. Majority vote wins."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "YES_BALLOTS = 6377482\n",
    "NO_BALLOTS = 6431376\n",
    "UNMARKED_BALLOTS = 86243\n",
    "NULL_BALLOTS = 170946"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "TOTAL_VOTES = YES_BALLOTS + NO_BALLOTS + UNMARKED_BALLOTS + NULL_BALLOTS\n",
    "P_INVALID = .02\n",
    "P_MISCLASSIFICATION = .01\n",
    "N_TRIALS = 100000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def simulate_vote(probability_yes):\n",
    "    yes_votes = int(TOTAL_VOTES * probability_yes)\n",
    "    no_votes = TOTAL_VOTES - yes_votes\n",
    "    \n",
    "    yes_votes_samples = N_TRIALS * [yes_votes]\n",
    "    no_votes_samples = N_TRIALS * [no_votes]\n",
    "    \n",
    "    invalid_ballots_yes = np.random.binomial(n=yes_votes_samples, p=P_INVALID)\n",
    "    invalid_ballots_no = np.random.binomial(n=no_votes_samples, p=P_INVALID)\n",
    "    \n",
    "    valid_yes_votes = yes_votes - invalid_ballots_yes\n",
    "    valid_no_votes = no_votes - invalid_ballots_no\n",
    "    \n",
    "    yes_votes_from_yes_voters = np.random.binomial(n=valid_yes_votes, p=1-P_MISCLASSIFICATION)\n",
    "    no_votes_from_yes_voters = valid_yes_votes - yes_votes_from_yes_voters\n",
    "    \n",
    "    no_votes_from_no_voters = np.random.binomial(n=valid_no_votes, p=1-P_MISCLASSIFICATION)\n",
    "    yes_votes_from_no_voters = valid_no_votes - no_votes_from_no_voters\n",
    "    \n",
    "    tallied_yes_votes = yes_votes_from_yes_voters + yes_votes_from_no_voters\n",
    "    tallied_no_votes = no_votes_from_no_voters + no_votes_from_yes_voters\n",
    "    \n",
    "    return tallied_yes_votes / (tallied_yes_votes + tallied_no_votes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p_yes: 60.000000% | no_win_percentage: 0.000%\n",
      "p_yes: 51.000000% | no_win_percentage: 0.000%\n",
      "p_yes: 50.100000% | no_win_percentage: 0.000%\n",
      "p_yes: 50.010000% | no_win_percentage: 0.191%\n",
      "p_yes: 50.001000% | no_win_percentage: 38.688%\n",
      "p_yes: 50.000100% | no_win_percentage: 48.791%\n",
      "p_yes: 50.000010% | no_win_percentage: 50.063%\n"
     ]
    }
   ],
   "source": [
    "for epsilon in [1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6, 1e-7]:\n",
    "    probability_yes = .5 + epsilon\n",
    "    percentage_of_tallied_votes_that_were_yes = simulate_vote(probability_yes)\n",
    "    proportion_of_trials_won_by_no = (percentage_of_tallied_votes_that_were_yes < .5).mean()\n",
    "    \n",
    "    results = \"p_yes: {:1.6f}% | no_win_percentage: {:1.3f}%\"\n",
    "    print(results.format(100*probability_yes, 100*proportion_of_trials_won_by_no))"
   ]
  }
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