{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# HIDDEN\n", "from datascience import *\n", "%matplotlib inline\n", "import matplotlib.pyplot as plots\n", "plots.style.use('fivethirtyeight')\n", "import math\n", "import numpy as np\n", "from scipy import stats" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# HIDDEN\n", "\n", "colors = Table.read_table('roulette_wheel.csv').column('Color')\n", "pockets = make_array('0','00')\n", "for i in np.arange(1, 37):\n", " pockets = np.append(pockets, str(i)) \n", "\n", "wheel = Table().with_columns(\n", " 'Pocket', pockets,\n", " 'Color', colors\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Central Limit Theorem ###\n", "Very few of the data histograms that we have seen in this course have been bell shaped. When we have come across a bell shaped distribution, it has almost invariably been an empirical histogram of a statistic based on a random sample.\n", "\n", "The examples below show two very different situations in which an approximate bell shape appears in such histograms." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Net Gain in Roulette ###\n", "In an earlier section, the bell appeared as the rough shape of the total amount of money we would make if we placed the same bet repeatedly on different spins of a roulette wheel. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "
Color | \n", "|
---|---|
0 | green | \n", "
00 | green | \n", "
1 | red | \n", "
2 | black | \n", "
3 | red | \n", "
4 | black | \n", "
5 | red | \n", "
6 | black | \n", "
7 | red | \n", "
8 | black | \n", "
... (28 rows omitted)
\n", " \n", "... (28 rows omitted)
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "red.select('Winnings: Red').hist(bins=np.arange(-1.5, 1.6, 1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now suppose you bet many times on red. Your net winnings will be the sum of many draws made at random with replacement from the distribution above.\n", "\n", "It will take a bit of math to list all the possible values of your net winnings along with all of their chances. We won't do that; instead, we will approximate the probability distribution by simulation, as we have done all along in this course. \n", "\n", "The code below simulates your net gain if you bet \\$1 on red on 400 different spins of the roulette wheel. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": true }, "outputs": [], "source": [ "num_bets = 400\n", "repetitions = 10000\n", "\n", "net_gain_red = make_array()\n", "\n", "for i in np.arange(repetitions):\n", " spins = red.sample(num_bets)\n", " new_net_gain_red = spins.column('Winnings: Red').sum()\n", " net_gain_red = np.append(net_gain_red, new_net_gain_red)\n", "\n", "\n", "results = Table().with_column(\n", " 'Net Gain on Red', net_gain_red\n", " )" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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BAUOHDoVUKlWs19VxVRozZgwkEgn27duntFyXxlVUVITw8HD069cPrVu3Rteu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\nkyr7f55UKoVEIqlyM46trS3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"text/plain": [ "Color | \n", "
---|
Purple | \n", "
Purple | \n", "
Purple | \n", "
White | \n", "