{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Part 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The effect of each bet on your capital is multiplicative - your capital is multiplied by $1 + 2f$ on heads and by $1 - f$ on tails - so that your fortune at the end depends only on the total number of heads and tails, not on the order in which they occured.\n", "\n", "In particular, for a given $f$, there will be a number $h(f)$ such that if the number of heads flipped is greater than or equal to $h(f)$, our final fortune will be greater than one billion. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "f = 0.25\n", "hs = range(1001)\n", "final_amounts = [(1 + 2 * f)**h * (1 - f)**(1000 - h) for h in hs]\n", "\n", "big_enough = [h for h, final_amount in zip(hs, final_amounts) if final_amount >= 10**9]\n", "\n", "fig, ax = plt.subplots(figsize=(12, 8))\n", "ax.semilogy(hs, final_amounts)\n", "ax.axhline(1e9, color='red', linestyle='--')\n", "ax.axvspan(min(big_enough), max(big_enough), color='green', alpha=0.2)\n", "ax.set_xlim(0, 1000)\n", "ax.set_ylabel('Final fortune')\n", "ax.set_xlabel('Number of heads')\n", "ax.set_title('{0} heads needed for f = {1}'.format(min(big_enough), f));" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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SCVJTU/Hdd99h6dKlGDp0KAIDA2FnZ4exY8fi3XffhbGxMXbv3g2VSlXl/AsXLkTv3r2h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"text": [ "" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can solve directly for the number of heads needed for a given $f$.\n", "\n", "$$ (1 + 2f)^h (1 - f)^{1000 - h} = 10^9$$\n", "\n", "$$ h \\log_{10} (1 + 2f) + (1000 - h) \\log_{10} (1 - f) = 9 $$\n", "\n", "$$ h = \\left \\lceil{\\frac{9 - 1000 \\log_{10} (1 - f)}{\\log_{10} (1 + 2f) - \\log_{10} (1 - f)}}\\right \\rceil $$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def min_h_greater_given_f(f, mult=2):\n", " ''' Returns the minimimum number of heads needed to end up with at least\n", " 10^9 pounds after 1000 flips if your capital is multiplied by \n", " (1 + mult * f) for each head and by (1 - f) for each tail. If even\n", " 1000 heads are not enough, returns 1001.\n", " '''\n", " if f == 0:\n", " # Impossible to end up above 1 if wagering 0.\n", " h = 1001\n", " elif f == 1:\n", " # Need to never get a tail if wagering everything.\n", " h = 1000\n", " else:\n", " exact = (9 - 1000 * np.log10(1 - f)) / (np.log10(1 + mult * f) - np.log10(1 - f))\n", " h = min(1001, np.ceil(exact))\n", " return h" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "An optimal $f$ minimizes the number of heads needed. Notice that because the number of heads is discrete, a range of $f$ values will be optimal. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "fs = np.linspace(0, 1, 1000)\n", "hs = [min_h_greater_given_f(f) for f in fs]\n", "\n", "min_h = min(hs)\n", "optimal_fs = [f for f, h in zip(fs, hs) if h == min_h]\n", "\n", "fig, (full_ax, zoom_ax) = plt.subplots(1, 2, figsize=(12, 6))\n", "\n", "full_ax.plot(fs, hs)\n", "full_ax.set_ylim(0, 1001)\n", "full_ax.set_xlabel('f')\n", "full_ax.set_ylabel('Heads needed')\n", "\n", "zoom_ax.plot(fs, hs)\n", "zoom_ax.set_ylim(min_h - 10, min_h + 10)\n", "zoom_ax.set_xlim(min(optimal_fs) - 0.1, max(optimal_fs) + 0.1);\n", "zoom_ax.set_xlabel('f')\n", "zoom_ax.set_ylabel('Heads needed');" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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FxM4++QR69bK6Cgl1CuPiDwrjvrNVGK+srGTcuHG0b9+eJk2a0L59e8aPH09VVZXXebm5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\nRk5OjmEYhlFdXW2kpqYaEydO9JxbXl5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AAAyhjAMAAACGUMYBAAAAQyjjAAAAgCGUcQAA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\n", "text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The probability of achieving a given number of heads is given by a Binomial(1000, 0.5) distribution. We are interested in the probability of getting greater than or equal to a given number of heads. We can compute this ourselves from the formula..." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.misc import comb\n", "\n", "hs = range(1002)\n", "binomial_coefficients = [comb(1000, h) for h in hs]\n", "pdf = np.array(binomial_coefficients) / 2.**1000\n", "p_geq = np.cumsum(pdf[::-1])[::-1]\n", "\n", "fig, (pdf_ax, p_geq_ax) = plt.subplots(1, 2, figsize=(12, 6))\n", "\n", "pdf_ax.plot(hs, pdf)\n", "pdf_ax.axis('tight')\n", "pdf_ax.set_ylabel('Probability that H = h')\n", "pdf_ax.set_xlabel('h')\n", "\n", "p_geq_ax.plot(hs, p_geq)\n", "p_geq_ax.set_ylabel('Probability that H >= h')\n", "p_geq_ax.set_xlabel('h')\n", "p_geq_ax.axis('tight');" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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FY/bv36/evXtrzJgxGjFihIqKivTMM89IkjZs2OBp4SlntnOGdXXsKP3lL9JVV4W6EoQL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\nAQBVC/rcsQULFrjatGnjiomJcfXo0cP14Ycfen4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"text": [ "" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "... or use `scipy`'s built-in binomial distribution object." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.stats import binom\n", "\n", "p_geq_alternate = [binom(1000, 0.5).sf(h - 1) for h in range(1002)]\n", "\n", "print np.allclose(p_geq, p_geq_alternate)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "True\n" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution is the probability of getting at least the minimum number of heads." ] }, { "cell_type": "code", "collapsed": false, "input": [ "print p_geq[min_h]" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.999992836186\n" ] } ], "prompt_number": 6 }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Part 2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The problem as stated multiplies your wager by 2 on a win. What is the smallest this factor can be while still leaving the probability of ending up above one billion greater than 0.5, assuming that you play with an optimal f given the value of the factor?\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def best_p_given_mult_crude(mult, return_f=False):\n", " ''' For a given value mult of the factor, returns the probability \n", " of ending up a billionaire given optimal f.\n", " '''\n", " fs = np.linspace(0, 1, 1000)\n", " min_h = min(min_h_greater_given_f(f, mult) for f in fs)\n", " best_p = p_geq[min_h]\n", " return best_p" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 7 }, { "cell_type": "code", "collapsed": false, "input": [ "fig, axs = plt.subplots(1, 2, figsize=(16, 8))\n", "lims_list = [(1, 2), (1.48, 1.52)]\n", "\n", "for ax, (x_min, x_max) in zip(axs, lims_list):\n", " mults = np.linspace(x_min, x_max, 1000)\n", " ps = [best_p_given_mult_crude(mult) for mult in mults]\n", "\n", " ax.plot(mults, ps)\n", "\n", " ax.axis('tight')\n", " ax.set_ylabel('P(billionaire) given optimal f')\n", " ax.set_xlabel('Factor on heads')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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JdjaWgAIQEIYNk3bv9s5qXaeO1dEAAACUjU6ys1EkAwgIixdL8+dLDRpIsbFWRwMAAFA2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\n8cYbGjdunCfDBgAAAABUEB69J3nAgAE6c+aMpk2bpszMTN1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"text": [ "" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We know that p < 0.5 if the factor is 1 and p > 0.5 if the factor is 2. We can repeatedly bisect this interval, maintaining the invariant that the left point of each new interval always produces a p < 0.5 and the right point always produces a p > 0.5. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "def perform_bisection(best_p_finder):\n", " low, high = 1., 2.\n", " num_digits = 15\n", " while (high - low) > 10**-num_digits:\n", " mid = (low + high) / 2\n", " mid_p = best_p_finder(mid)\n", " if mid_p < 0.5:\n", " low = mid\n", " else:\n", " high = mid\n", " \n", " print 'A factor of {0:0.{precision}f} is high enough, but {1:0.{precision}f} is not.'.format(high, low, precision=num_digits + 1)\n", " return high" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "min_factor = perform_bisection(best_p_given_mult_crude)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "A factor of 1.5046762137696037 is high enough, but 1.5046762137696028 is not.\n" ] } ], "prompt_number": 10 }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "I was wrong!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As group 2 has pointed out, the above is wrong in the sense that it is not as small as possible to the claimed number of digits. The problem is that the method for finding the optimal $f$ was too crude. I was checking 1000 equally spaced points between 0 and 1. If the range of optimal $f$'s is small enough to fall entirely between these points, it won't be found, and the reported minimum number of heads will be too high. For the smaller value of the factor they report, 100,000 points are enough to show this." ] }, { "cell_type": "code", "collapsed": false, "input": [ "smaller_mult = 1.504673877089423\n", "\n", "fig, axs = plt.subplots(1, 2, figsize=(12, 6))\n", "num_points_list = [1000, 100000]\n", "\n", "for ax, num_points in zip(axs, num_points_list):\n", " fs = np.linspace(0, 1, num_points)\n", " hs = [min_h_greater_given_f(f, mult=smaller_mult) for f in fs]\n", "\n", " min_h = min(hs)\n", " optimal_fs = [f for f, h in zip(fs, hs) if h == min_h]\n", "\n", " ax.plot(fs, hs)\n", " ax.axvspan(min(optimal_fs), max(optimal_fs), color='green', alpha=0.2)\n", " ax.set_ylim(min_h - 10, min_h + 10)\n", " ax.set_xlim(0.05, 0.3);\n", " ax.set_xlabel('f')\n", " ax.set_title('{0:,} points'.format(num_points))\n", " ax.set_ylabel('Heads needed');" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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Uti5yTNu3A+fPN37d9es8mNSe+DcRmcyeDXTsCJw4ASxZInU1pBY5OUBmJlBc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\n49SpU1iwYAFOnjyJ7Oxse+4KEREREZFN2S2onzt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"text": [ "" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course, if 1,000 points wasn't enough, how do we know 1,000,000 is? In retrospect, it is clear that as you approach the infimum of all valid factors from the right, the range of optimal $f$'s will become arbitrarily small, so for any number of equally spaced points, we can get close enough to the infimum to make the range of optimal $f$'s fall through the cracks. \n", "\n", "To get around this, we can do more sophisticated numerical minimization of the equation for $h$ in terms of $f$. Since this requires $h$ to be a continuous function of $f$, we need to do this while $h$ is still allowed to take fractional values, then take the `ceil` *after* minimizing. (This was John's idea.)" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.optimize import minimize_scalar\n", "\n", "def best_p_given_mult(mult, return_f=False):\n", " ''' For a given value mult of the factor, returns the probability \n", " of ending up a billionaire given optimal f.\n", " '''\n", " get_fractional_h = lambda f: (9 - 1000 * np.log10(1 - f)) / (np.log10(1 + mult * f) - np.log10(1 - f))\n", " minimization_result = minimize_scalar(get_fractional_h,\n", " bounds=(0.001, 0.999),\n", " method='Bounded',\n", " tol=1e-16,\n", " )\n", " fractional_h = minimization_result.fun\n", " min_h = min(1001, np.ceil(fractional_h))\n", " best_p = p_geq[min_h]\n", " if return_f:\n", " return best_p, minimization_result.x\n", " else:\n", " return best_p" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 12 }, { "cell_type": "code", "collapsed": false, "input": [ "even_smaller_mult = perform_bisection(best_p_given_mult)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "A factor of 1.5046738770873436 is high enough, but 1.5046738770873427 is not.\n" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is smaller than the number reported by group 2 by more than $10^{-15}$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "smaller_mult - even_smaller_mult" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 14, "text": [ "2.079447725122918e-12" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's confirm that there exists an $f$ that makes this a valid factor." ] }, { "cell_type": "code", "collapsed": false, "input": [ "p, optimal_f = best_p_given_mult(even_smaller_mult, return_f=True)\n", "min_h_greater_given_f(optimal_f, even_smaller_mult)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 15, "text": [ "500.0" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "So I'm back on top!" ] } ], "metadata": {} } ] }