{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical simulations of the Monty-Hall \"paradox\"\n",
    "This short notebook aims at simulating trials of the so-called [Monty-Hall problem](https://en.wikipedia.org/wiki/Monty_Hall_problem), and thus helping to convince about the result thanks to a numerical results rather than a possibly-unclear proof."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Definition of the problem\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is $M \\geq 3$ doors, and behind only one of them there is a treasure.\n",
    "The goal of the player is to find the treasure, following this game:\n",
    "\n",
    "1. The player first chose a door, but does not open it yet,\n",
    "2. All remaining doors but one are opened, and none of them contain the treasure. The player sees $M - 2$ bad doors,\n",
    "3. So there is just 2 doors, the one first chosen, and the last one. She knows the treasure is behind one of them,\n",
    "4. And she has to decide if she wants to **stay on her initial choice**, or **switches to the last door**.\n",
    "\n",
    "Finally, the chosen door is opened, and the player wins this round if she found the treasure."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "The goal of this notebook is to numerically prove that the choice of always switching to the last door is the best one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Starting our numerical simulations\n",
    "We start by importing some modules, then define a function, `randomAllocation()`, to generate a random allocation of treasures behind the `M` doors.\n",
    "\n",
    "> *Note:* all the code below is generic for any $M \\geq 3$, but `M = 3` is used to keep small and clear visualizations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = 3\n",
    "allocation = [False] * (M - 1) + [True]  # Only 1 treasure!\n",
    "assert set(allocation) == {True, False}  # Check: only True and False\n",
    "assert sum(allocation) == 1              # Check: only 1 treasure!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just to check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[False, False, True]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "allocation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can generate a random spot for the treasure by simply shuffling (with [`random.shuffle()`](https://docs.python.org/3/library/random.html#random.shuffle)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def randomAllocation():\n",
    "    r = allocation[:]\n",
    "    random.shuffle(r)\n",
    "    return r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us quickly check this function `randomAllocation()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False, True, False]\n",
      "[False, False, True]\n",
      "[False, False, True]\n",
      "[True, False, False]\n",
      "[False, False, True]\n",
      "[False, True, False]\n",
      "[True, False, False]\n",
      "[False, True, False]\n",
      "[False, True, False]\n",
      "[True, False, False]\n"
     ]
    }
   ],
   "source": [
    "for _ in range(10):\n",
    "    print(randomAllocation())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need to write a small function to simulate the choice of the door to show to the player, `show()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def last(r, i):\n",
    "    # Select a random index corresponding of the door we keep\n",
    "    if r[i]:  # She found the treasure, returning a random last door\n",
    "        return random.choice([j for (j, v) in enumerate(r) if j != i])\n",
    "    else:     # She didn't find the treasure, returning the treasure door\n",
    "        # Indeed, the game only removes door that don't contain the treasure\n",
    "        return random.choice([j for (j, v) in enumerate(r) if j != i and v])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- r = [False, False, True] i = 0 and last(r, i) = 2\n",
      "  Stay on False or go to True ?\n",
      "- r = [True, False, False] i = 1 and last(r, i) = 0\n",
      "  Stay on False or go to True ?\n",
      "- r = [False, False, True] i = 1 and last(r, i) = 2\n",
      "  Stay on False or go to True ?\n",
      "- r = [True, False, False] i = 1 and last(r, i) = 0\n",
      "  Stay on False or go to True ?\n",
      "- r = [False, False, True] i = 1 and last(r, i) = 2\n",
      "  Stay on False or go to True ?\n",
      "- r = [False, False, True] i = 0 and last(r, i) = 2\n",
      "  Stay on False or go to True ?\n",
      "- r = [False, True, False] i = 2 and last(r, i) = 1\n",
      "  Stay on False or go to True ?\n",
      "- r = [True, False, False] i = 1 and last(r, i) = 0\n",
      "  Stay on False or go to True ?\n",
      "- r = [False, False, True] i = 0 and last(r, i) = 2\n",
      "  Stay on False or go to True ?\n",
      "- r = [False, True, False] i = 1 and last(r, i) = 2\n",
      "  Stay on True or go to False ?\n"
     ]
    }
   ],
   "source": [
    "for _ in range(10):\n",
    "    r = randomAllocation()\n",
    "    i = random.randint(0, M - 1)\n",
    "    j = last(r, i)\n",
    "    print(\"- r =\", r, \"i =\", i, \"and last(r, i) =\", j)\n",
    "    print(\"  Stay on\", r[i], \"or go to\", r[j], \"?\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need a function to simulate the first choice of the player, and a simple choice is to select a uniform choice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def firstChoice():\n",
    "    global M\n",
    "    # Uniform first choice\n",
    "    return random.randint(0, M - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can simulate a game, for a certain left-to-be-written function `strategy()` that decides to keep or to change the initial choice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def simulate(stayOrNot):\n",
    "    # Random spot for the treasure\n",
    "    r = randomAllocation()\n",
    "    # Initial choice\n",
    "    i = firstChoice()\n",
    "    # Which door are remove, or equivalently which is the last one to be there?\n",
    "    j = last(r, i)\n",
    "    assert {r[i], r[j]} == {False, True}  # There is still the treasure and only one\n",
    "    stay = stayOrNot()\n",
    "    if stay:\n",
    "        return r[i]\n",
    "    else:\n",
    "        return r[j]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can simulate many outcome of the game for one strategy, and return the number of time it won (i.e. average number of time it found the good door, by finding `r[i] = True` or `r[j] = True`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "N = 10000\n",
    "\n",
    "def simulateManyGames(stayOrNot):\n",
    "    global N\n",
    "    results = [simulate(stayOrNot) for _ in range(N)]\n",
    "    return sum(results)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Comparing two strategies, on many randomized trials\n",
    "We will simulate the two strategies, `keep()` vs `change()`, on $N = 10000$ randomized games."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Keeping our first choice, `keep()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def keep():\n",
    "    return True  # True == also stay on our first choice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- For 10000 simulations, the strategy 'keep' has won 3280 of the trials...\n",
      "  ==> proportion = 32.80%.\n"
     ]
    }
   ],
   "source": [
    "rate = simulateManyGames(keep)\n",
    "print(\"- For\", N, \"simulations, the strategy 'keep' has won\", rate, \"of the trials...\")\n",
    "print(\"  ==> proportion = {:.2%}.\".format(rate / float(N)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\implies$ We find a chance of winning of about $\\frac{1}{M} = \\frac{1}{3}$ for this strategy, which is very logical as only the initial choice matters, and due to the uniform location of the treasure behind the $M = 3$ doors, and the uniform first choice with `firstChoice()`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Changing our first choice, `change()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def change():\n",
    "    return False  # False == never stay, ie always chose the last door"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- For 10000 simulations, the strategy 'change' has won 6693  of the trials...\n",
      "  ==> proportion = 66.93%.\n"
     ]
    }
   ],
   "source": [
    "rate = simulateManyGames(change)\n",
    "print(\"- For\", N, \"simulations, the strategy 'change' has won\", rate, \" of the trials...\")\n",
    "print(\"  ==> proportion = {:.2%}.\".format(rate / float(N)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\implies$ We find a chance of winning of about $\\frac{M - 1}{M} = \\frac{2}{3}$ for this strategy, which is less logical.\n",
    "\n",
    "Due to the uniform location of the treasure behind the $M = 3$ doors, and the uniform first choice with `firstChoice()`, we have a $\\frac{1}{M}$ chance of finding the treasure from the first time.\n",
    "\n",
    "1. If we found it, the last door does not contain the treasure, hence **we loose** as we switch to it,\n",
    "2. However, if we did not find it, the last door has to contain the treasure, hence **we win** as we switch to it deterministically (i.e. always).\n",
    "\n",
    "The first case has probability $\\frac{1}{M}$, probability of loosing, and the second case has probability $\\frac{M - 1}{M}$, probability of loosing.\n",
    "\n",
    "$\\implies$ Conclusion : this strategy `change()` has a chance of winning of $\\frac{2}{3}$, far better than the chance of $\\frac{1}{3}$ for `stay()`.\n",
    "\n",
    "> We proved numerically the results given and explained [here on the Wikipedia page](https://en.wikipedia.org/wiki/Monty_Hall_problem#Direct_calculation). Great!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bernoulli choice\n",
    "We can try a randomized strategy, a simple one can be to follow the decision of a (biased) coin:\n",
    "\n",
    "1. Toss a coin (with fixed head probability equals to $p \\in [0, 1]$)\n",
    "2. If head, switch to the last door."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def bernoulli(p=0.5):\n",
    "    return random.random() < p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- For 10000 simulations, the strategy 'bernoulli' has won 5077  of the trials...\n",
      "  ==> proportion = 50.77%.\n"
     ]
    }
   ],
   "source": [
    "rate = simulateManyGames(bernoulli)\n",
    "print(\"- For\", N, \"simulations, the strategy 'bernoulli' has won\", rate, \" of the trials...\")\n",
    "print(\"  ==> proportion = {:.2%}.\".format(rate / float(N)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can try different values for $p$, and plot the resulting chance of winning the game as a function of $p$.\n",
    "Hopefully, it should be monotonic, confirming the result explained above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We generate lots of values for $p$, then a function `stratBernoulli()` to create the strategy described above, for some $p \\in [0, 1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "values_p = np.linspace(0, 1, 500)\n",
    "\n",
    "def stratBernoulli(p):\n",
    "    def stayOrNot():\n",
    "        return bernoulli(p=p)\n",
    "    return stayOrNot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let finally do all the simulations, and store the empirical probability of winning the game when following a Bernoulli strategy.\n",
    "\n",
    "> This line takes about $4$ minutes on my laptop, it's not that quick."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "chance_of_winning = [simulateManyGames(stratBernoulli(p)) / float(N) for p in values_p]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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nTjPbw8zam9lKZrYHcAcwJFlnI+Dj2iqkiIiILFwKCVCOAp4HHgK+AL5Mbj8P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\no3WriWkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6558b82b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(values_p, chance_of_winning, 'r')\n",
    "plt.title(\"Monty-Hall paradox with {} doors ({} random simulation)\".format(M, N))\n",
    "plt.xlabel(\"Probability $p$ of staying on our first choice (Bernoulli strategy)\")\n",
    "plt.ylabel(\"Probability of winning\")\n",
    "plt.ylim(0, 1)\n",
    "plt.yticks(np.linspace(0, 1, 11))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## Examples with $M = 100$ doors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def completeSimu():\n",
    "    global M\n",
    "    global N\n",
    "    allocation = [False] * (M - 1) + [True]  # Only 1 treasure!\n",
    "    \n",
    "    def randomAllocation():\n",
    "        r = allocation[:]\n",
    "        random.shuffle(r)\n",
    "        return r\n",
    "    \n",
    "    def last(r, i):\n",
    "        # Select a random index corresponding of the door we keep\n",
    "        if r[i]:  # She found the treasure, returning a random last door\n",
    "            return random.choice([j for (j, v) in enumerate(r) if j != i])\n",
    "        else:     # She didn't find the treasure, returning the treasure door\n",
    "            # Indeed, the game only removes door that don't contain the treasure\n",
    "            return random.choice([j for (j, v) in enumerate(r) if j != i and v])\n",
    "    \n",
    "    def simulate(stayOrNot):\n",
    "        # Random spot for the treasure\n",
    "        r = randomAllocation()\n",
    "        # Initial choice\n",
    "        i = firstChoice()\n",
    "        # Which door are remove, or equivalently which is the last one to be there?\n",
    "        j = last(r, i)\n",
    "        stay = stayOrNot()\n",
    "        if stay:\n",
    "            return r[i]\n",
    "        else:\n",
    "            return r[j]\n",
    "\n",
    "    def simulateManyGames(stayOrNot):\n",
    "        global N\n",
    "        results = [simulate(stayOrNot) for _ in range(N)]\n",
    "        return sum(results)\n",
    "\n",
    "    values_p = np.linspace(0, 1, 300)\n",
    "    chance_of_winning = [simulateManyGames(stratBernoulli(p)) / float(N) for p in values_p]\n",
    "    plt.figure()\n",
    "    plt.plot(values_p, chance_of_winning, 'r')\n",
    "    plt.title(\"Monty-Hall paradox with {} doors ({} random simulation)\".format(M, N))\n",
    "    plt.xlabel(\"Probability $p$ of staying on our first choice (Bernoulli strategy)\")\n",
    "    plt.ylabel(\"Probability of winning\")\n",
    "    plt.ylim(0, 1)\n",
    "    plt.yticks(np.linspace(0, 1, 11))\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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T1+DMYczZzp1syyHy0nyTNMPWK3d/1t1PdfcNiQ69OwDb1+MhvwewBZMz/imPbfcGPnb3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\nWeE4QTEzM7PCcYJiZmZmhfO/3G4E6LXpd/AAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f655671a940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = 4\n",
    "completeSimu()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Last plot, for $M = 100$\n",
    "It clearly shows the linear behavior we expected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cm5rvxutTRuZhIke+JZU3QJsIrJql1CM1GuAXVF/OH8Wk+O8+4ICkSG4f4L6k+iHlEBZU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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f65565dbf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = 100\n",
    "completeSimu()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\implies$ it is now clear that the best strategy is the **always change and select the other door**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Conclusion\n",
    "As we saw it, it is clear from the numerical experiments presented above that the strategy of **always changing our first choice** (`change()`), is statistically better than the strategy of **keeping our first choice** (`keep()`).\n",
    "\n",
    "> [See this repository for other Python notebook doing numerical simulations](https://github.com/Naereen/notebooks/tree/master/simus/)."
   ]
  }
 ],
 "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": "3.5.2+"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}