Coin+toss+problem.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Problem posed by Alexander Bogomonly (https://twitter.com/CutTheKnotMath)\n",
    "\n",
    "Problem: A tosses a fair coin 20 times. B does that 21 times. \n",
    "\n",
    "What's the probability B will have more heads than A ?\n",
    "\n",
    "See:\n",
    "1. [Problem](https://twitter.com/CutTheKnotMath/status/924701881292685312)\n",
    "2. [Solutions](https://www.cut-the-knot.org/m/Probability/CoinTossingContest.shtml)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A combinatorial identity proposed by Taleb(https://twitter.com/nntaleb)\n",
    "\n",
    "[Combinatorial Identity](https://twitter.com/nntaleb/status/924805829387309056)\n",
    "\n",
    "\n",
    "Let \n",
    "m = number of cointosses by B\n",
    "\n",
    "n = number of cointosses by A\n",
    "\n",
    "Set p=q=1/2 (for a fair coin),\n",
    "\n",
    "which vastly simplifies Taleb's original result.\n",
    "\n",
    "For a fair coin, \n",
    "the probability P(B-A >=d), parametrized on the difference d between the number of heads, \n",
    "is then given by\n",
    "\n",
    "---------------------------------------------------------------------------------\n",
    "\n",
    "P(B-A >= d) = sum(w = d..max(m,n), (m choose w) x (1/2)^(m+n) x 2F1(-n,w-m,w+1,1))\n",
    "\n",
    "---------------------------------------------------------------------------------\n",
    "\n",
    "where 2F1(...) is the [generalized hypergeometric function](https://en.wikipedia.org/wiki/Generalized_hypergeometric_function#The_series_2F1)\n",
    "\n",
    "This probability is computed below for various game configurations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy.special as ss\n",
    "import fractions as ff\n",
    "\n",
    "def talebcointoss(m,n,d):\n",
    "    sum = 0\n",
    "    for w in range(d,max(m,n)+1):\n",
    "        s = ff.Fraction(1/2) ** (m+n)\n",
    "        c = ss.comb(m,w)\n",
    "        h = ss.hyp2f1(-n,w-m,w+1,1)\n",
    "        sum += s*c*h\n",
    "    return sum    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m=21\n",
    "n=20\n",
    "d=1\n",
    "talebcointoss(m,n,d) #Result for the original problem posed by Alexander"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Result obtained by Amit for m=22, n=20 coin tosses: https://twitter.com/gcfr20/status/924811626179629056"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.56119283562384226"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "talebcointoss(22,20,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Result obtained by Amit for m=3, n=1 coin tosses:https://twitter.com/gcfr20/status/924807489404710912"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6875"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "talebcointoss(3,1,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's visualize this probability cadlag as the number of tosses for player B increase, holding A fixed at 20."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DoP1SF3teaODMsV9ypO5H9HR3AdB24TyvbHmW7CbjuqN+XCg8vre1i9btJ/i/\nBa/1B3tUKBSirq6Oso/G/8zSlW4r/9P3yKClmbwsY32pPgZPRMZvLLtl7gROOucanXPdQA2wYsiY\nFcDzke9/DFSZWewawzUYbkvg7ENPDnul5nC3v/Vv2dYf7FGus5ODr3f2B3tUT3cf//izbf3B/l69\nC3eooz/Y+58n1Mfljva4PQSDQXJz4od1Vc4pnpo/m1k5fgBm5fh5av5sHiiOPZoXERmNOedGHmD2\nILDMOffVyOMvAYudc2sGjPlVZMyZyONTkTEXhjzXamA1wIwZM26vqakZ8bXb29spKCgA4EhTMO6Y\nj9jb8ZaqAXhzypS49dKW+D23TX1/3Hpfz6/j1qflxL+Q6JK102exr+Hz+XjftBw6O5vADfilYFnk\n5s7Enx0ABvedKTKxZ8jMvjOxZ0hc35WVlQedc+WjjUvqCVXn3BZgC0B5ebmrqKgYcfzevXuJjnns\nOz+nKc6e71/mPksx52P/cmA2fzz7JpqvxC6D/OVWx7Rgb0x9/z3fptNfGFPvbvs5fT2XY+oLP/A1\nrsuK/ZfVWBBin705aGnG7/dTXV3df1J16G6ZkuL3/mdoYN+ZIhN7hszsOxN7huT3PZZlmSZg9oDH\nsyK1uGPMLBsIED6xmjCPLp1Pnt83qJbn93F60aPDXqk53O1vQ6tXYrmD65aby+135JI9ZfA/kuwp\nWXz0UyvJnpIzpJ6DLcrD/IPHmz+L25ffQ3V1NYFA+Eg8EAj0BztASfEK7rlnH1X3nuSee/YNCnYR\nkUQYy5H768A8M7uZcIivAr4wZMzLwO8BrwEPAj93o633jFN0t8jQ3TJ3LFwGc94Xd7dM9L6IQ3fL\nVJQuJ1i0MM5umc9w3TC7ZUrmBmJ2y8xbUhF3t0z+wiLKKOoPcxGRZBs13J1zPWa2BthNeCvkD5xz\nR81sE/CGc+5l4PvAj8zsJHCJ8C+AhBt2S2DZymGv1Fxeujzu7W8D1dUEqqtj6rcsLuaWxbFr6QuW\nVLJgSWVMPX9hEfkLi8YwexGR5BnTmrtzbhewa0htw4DvO4F/ldipiYjI1dKNw0REPEjhLiLiQQp3\nEREPUriLiHiQwl1ExIMU7iIiHqRwFxHxoFFvHDZhL2x2HvinUYZNBy6MMsaLMrHvTOwZMrPvTOwZ\nEtf3B5xzN442KGXhPhZm9sZY7n7mNZnYdyb2DJnZdyb2DMnvW8syIiIepHAXEfGgdA/3LameQIpk\nYt+Z2DMpEB2IAAADZUlEQVRkZt+Z2DMkue+0XnMXEZGrk+5H7iIichXSNtzNbJmZHTezk2b2jVTP\nZ6KY2Q/M7Fzkc2ijtWlm9jMzOxH5+r5UzjHRzGy2me0xszfN7KiZrY3UPdu3meWa2f8zs3+M9Pxf\nIvWbzexA5H3+v8ws/gf/TmJm5jOzw2b208jjTOj5HTM7Ymb/YGZvRGpJfX+nZbibmQ94Dvg0cBvw\neTO7LbWzmjA/BJYNqX0DqHPOzQPqIo+9pAf4Q+fcbcDHga9F/v16ue8u4F7n3EeBjwHLzOzjwJ8A\nTzvnPgi8C3wlhXOcKGuBYwMeZ0LPAJXOuY8N2P6Y1Pd3WoY7cCdw0jnX6JzrBmoAT37QqHPuF4Q/\nvWqgFcDzke+fB+5P6qQmmHOu2Tl3KPJ9G+H/8Gfi4b5dWHvkoT/yxwH3Aj+O1D3VM4CZzQKWA9+L\nPDY83vMIkvr+TtdwnwmcHvD4TKSWKWY455oj37cAM1I5mYlkZnOAhcABPN53ZHniH4BzwM+AU0Cr\nc64nMsSL7/NngP8I9EUe34D3e4bwL+5XzOygma2O1JL6/h7Tx+xJ6jjnnJl5ckuTmRUALwIPO+cu\nhw/qwrzYt3OuF/iYmRUCPwFuTfGUJpSZ/TZwzjl30MwqUj2fJPuEc67JzIqAn5lZw8AfJuP9na5H\n7k3A7AGPZ0VqmeLXZlYCEPl6LsXzSTgz8xMO9hecc9sjZc/3DeCcawX2AHcBhWYWPcjy2vv8HuCz\nZvYO4aXVe4HNeLtnAJxzTZGv5wj/Ir+TJL+/0zXcXwfmRc6qTwFWAS+neE7J9DLwe5Hvfw/YkcK5\nJFxk3fX7wDHn3J8N+JFn+zazGyNH7JhZHvApwuca9gAPRoZ5qmfn3Hrn3Czn3BzC/w3/3Dn3RTzc\nM4CZ5ZvZ1Oj3wH3Ar0jy+zttL2Iys88QXq/zAT9wzn07xVOaEGb2N0AF4TvG/Rr4FvASsA14P+E7\nZ650zg096TppmdkngH3AEd5bi/0m4XV3T/ZtZmWET6L5CB9UbXPObTKzUsJHtdOAw8DvOue6UjfT\niRFZllnnnPttr/cc6e8nkYfZwF87575tZjeQxPd32oa7iIhcvXRdlhERkWugcBcR8SCFu4iIBync\nRUQ8SOEuIuJBCncREQ9SuIuIeJDCXUTEg/4/ik+V96xGOUMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7faf366dde10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.figure()\n",
    "plt.grid(b=True, which='both', axis='both')\n",
    "for m in range(1,51):\n",
    "    plt.scatter(m, talebcointoss(m,20,d))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some insights:\n",
    "\n",
    "1. Even when both players toss a fair coin an equal number of times (m=n=20), there's a 40% chance that one of them ends up with more heads than the other. Equal distributions do not collapse to zero.\n",
    "\n",
    "See: https://twitter.com/qwob/status/924796354135777282\n",
    "\n",
    "2. When the other player is given twice as many turns (m=40,n=20),its almost certain he has more heads( ~100%)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try another experiment.\n",
    "What if both players have the same number (m=n=20) of tosses, but we count the difference of heads between them.\n",
    "Suppose we plot the probability as a function of the difference of heads, we obtain -"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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0meffkHv6zFlct+HTBVWkducJVakFhk6aerWMJgvDXWqR9193vWGuScNlGUkqIcNdkkrI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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7faf30e9be10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.grid(b=True, which='both', axis='both')\n",
    "for d in range(0,10):\n",
    "        plt.scatter(d, talebcointoss(20,20,d))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We note that\n",
    "\n",
    "1. Given 2 players toss a fair coin the same number of times, there's a 44% chance one of them has atleast one more head than the other. So a fairly high chance of a 1-head lead.\n",
    "\n",
    "2. As we insist on a lead of more than 1 head, the probability drops off linearly.\n",
    "   Chances of a 2-head lead are about 30%, 4-head lead about 15%, 6-head about 5%, ~zero chances of an 8-head lead.\n",
    "   \n",
    "Now, you might hypothesize the probability is so low because the players are evenly matched.\n",
    "\n",
    "Lets let one player toss the coin twice as much as the other.\n",
    "\n",
    "m=40, n=20\n",
    "\n",
    "That looks like -"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7faf30989b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.grid(b=True, which='both', axis='both')\n",
    "for d in range(0,20):\n",
    "        plt.scatter(d, talebcointoss(40,20,d))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ha! If one player tosses the fair coin twice as much as the other,\n",
    "he's almost surely guaranteed to lead by 1 head.\n",
    "\n",
    "In fact, chances of leading the other player by 4 heads is 95% !\n",
    "\n",
    "Chances of a  10-head lead is an even 50%.\n",
    "\n",
    "In fact, a full 15 head lead, out of the 20 extra coin tosses, is a not insignificant 10% chance.\n",
    "\n",
    "So handicapping one player at the expense of another vastly boosts the latter's chances.\n",
    "\n",
    "We could study the entire landscape now, varying both player tosses and head difference between players -"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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r1gp+3sTc00pbl9tqKkoxyCv4xphRYCOwFzgMPGqMOSQi94vIpyfbwVLETWsF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\nmGVXrWz2FNyEdAhl+UDLZk8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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7faf308a7a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.grid(b=True, which='both', axis='both')\n",
    "for m in range(20,40):\n",
    "    for d in range(0,20):\n",
    "        plt.scatter(d, talebcointoss(m,20,d))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see the previous 2 curves enveloping the landscape of possibilities.\n",
    "\n",
    "The uppermost curve is the 40-20 toss curve, the lowermost is the evenly matched 20-20 toss curve.\n",
    "\n",
    "In between the two are a a wide range of possibilities."
   ]
  }
 ],
 "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
}