Law of large numbers.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Law of large numbers (LLN)\n",
    "Average of a large number of experiments = Expected value\n",
    "\n",
    "\n",
    "This law has an application in the game roulette. A casino may lose or gain money in a single spin of roulette whell but overtime, after a large number of spins the profit will tend towards a predictable value. Unless the game is rigged, you will always make a predicted profit by opening a casino.\n",
    "\n",
    "\n",
    "So, for a dice roll the expected value of a single roll is:\n",
    "\n",
    "\n",
    "1 + 2 + 3... + 6 / 6 = 3.5\n",
    "\n",
    "\n",
    "so, according to LLN average of a large number of die roll shall be equal to 3.5. \n",
    "\n",
    "Let's test it empirically for ten thousand observations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Dice():\n",
    "    def __init__(self):\n",
    "        self.outcome = None\n",
    "    \n",
    "    def roll(self, n=1):\n",
    "        self.outcome = np.random.randint(1, 7, n)\n",
    "        return self.outcome\n",
    "    \n",
    "    def reset(self):\n",
    "        self.outcome= None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "MAX_N = int(1e4)\n",
    "dice = Dice()\n",
    "averages = []\n",
    "for n in range(1, MAX_N):\n",
    "    rolls = dice.roll(n)\n",
    "    average = np.average(rolls)\n",
    "    averages.append(average)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7ff45d305a58>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(averages, label='Expectation')\n",
    "plt.plot([3.5 for x in range(MAX_N)], label='Average')\n",
    "plt.legend(loc='upper right')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusion\n",
    "Clearly the expectation (blue line) in the graph above converges the yellow line (average)"
   ]
  }
 ],
 "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}