7.Probability & Stats 15th Jan.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "coin_tosses = np.random.binomial(n=1, p=.5, size=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.count_nonzero(coin_tosses)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "var_coin_tosses = np.arange(start=10, stop=100000, step=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([   10,   110,   210,   310,   410,   510,   610,   710,   810,\n",
       "         910,  1010,  1110,  1210,  1310,  1410,  1510,  1610,  1710,\n",
       "        1810,  1910,  2010,  2110,  2210,  2310,  2410,  2510,  2610,\n",
       "        2710,  2810,  2910,  3010,  3110,  3210,  3310,  3410,  3510,\n",
       "        3610,  3710,  3810,  3910,  4010,  4110,  4210,  4310,  4410,\n",
       "        4510,  4610,  4710,  4810,  4910,  5010,  5110,  5210,  5310,\n",
       "        5410,  5510,  5610,  5710,  5810,  5910,  6010,  6110,  6210,\n",
       "        6310,  6410,  6510,  6610,  6710,  6810,  6910,  7010,  7110,\n",
       "        7210,  7310,  7410,  7510,  7610,  7710,  7810,  7910,  8010,\n",
       "        8110,  8210,  8310,  8410,  8510,  8610,  8710,  8810,  8910,\n",
       "        9010,  9110,  9210,  9310,  9410,  9510,  9610,  9710,  9810,\n",
       "        9910, 10010, 10110, 10210, 10310, 10410, 10510, 10610, 10710,\n",
       "       10810, 10910, 11010, 11110, 11210, 11310, 11410, 11510, 11610,\n",
       "       11710, 11810, 11910, 12010, 12110, 12210, 12310, 12410, 12510,\n",
       "       12610, 12710, 12810, 12910, 13010, 13110, 13210, 13310, 13410,\n",
       "       13510, 13610, 13710, 13810, 13910, 14010, 14110, 14210, 14310,\n",
       "       14410, 14510, 14610, 14710, 14810, 14910, 15010, 15110, 15210,\n",
       "       15310, 15410, 15510, 15610, 15710, 15810, 15910, 16010, 16110,\n",
       "       16210, 16310, 16410, 16510, 16610, 16710, 16810, 16910, 17010,\n",
       "       17110, 17210, 17310, 17410, 17510, 17610, 17710, 17810, 17910,\n",
       "       18010, 18110, 18210, 18310, 18410, 18510, 18610, 18710, 18810,\n",
       "       18910, 19010, 19110, 19210, 19310, 19410, 19510, 19610, 19710,\n",
       "       19810, 19910, 20010, 20110, 20210, 20310, 20410, 20510, 20610,\n",
       "       20710, 20810, 20910, 21010, 21110, 21210, 21310, 21410, 21510,\n",
       "       21610, 21710, 21810, 21910, 22010, 22110, 22210, 22310, 22410,\n",
       "       22510, 22610, 22710, 22810, 22910, 23010, 23110, 23210, 23310,\n",
       "       23410, 23510, 23610, 23710, 23810, 23910, 24010, 24110, 24210,\n",
       "       24310, 24410, 24510, 24610, 24710, 24810, 24910, 25010, 25110,\n",
       "       25210, 25310, 25410, 25510, 25610, 25710, 25810, 25910, 26010,\n",
       "       26110, 26210, 26310, 26410, 26510, 26610, 26710, 26810, 26910,\n",
       "       27010, 27110, 27210, 27310, 27410, 27510, 27610, 27710, 27810,\n",
       "       27910, 28010, 28110, 28210, 28310, 28410, 28510, 28610, 28710,\n",
       "       28810, 28910, 29010, 29110, 29210, 29310, 29410, 29510, 29610,\n",
       "       29710, 29810, 29910, 30010, 30110, 30210, 30310, 30410, 30510,\n",
       "       30610, 30710, 30810, 30910, 31010, 31110, 31210, 31310, 31410,\n",
       "       31510, 31610, 31710, 31810, 31910, 32010, 32110, 32210, 32310,\n",
       "       32410, 32510, 32610, 32710, 32810, 32910, 33010, 33110, 33210,\n",
       "       33310, 33410, 33510, 33610, 33710, 33810, 33910, 34010, 34110,\n",
       "       34210, 34310, 34410, 34510, 34610, 34710, 34810, 34910, 35010,\n",
       "       35110, 35210, 35310, 35410, 35510, 35610, 35710, 35810, 35910,\n",
       "       36010, 36110, 36210, 36310, 36410, 36510, 36610, 36710, 36810,\n",
       "       36910, 37010, 37110, 37210, 37310, 37410, 37510, 37610, 37710,\n",
       "       37810, 37910, 38010, 38110, 38210, 38310, 38410, 38510, 38610,\n",
       "       38710, 38810, 38910, 39010, 39110, 39210, 39310, 39410, 39510,\n",
       "       39610, 39710, 39810, 39910, 40010, 40110, 40210, 40310, 40410,\n",
       "       40510, 40610, 40710, 40810, 40910, 41010, 41110, 41210, 41310,\n",
       "       41410, 41510, 41610, 41710, 41810, 41910, 42010, 42110, 42210,\n",
       "       42310, 42410, 42510, 42610, 42710, 42810, 42910, 43010, 43110,\n",
       "       43210, 43310, 43410, 43510, 43610, 43710, 43810, 43910, 44010,\n",
       "       44110, 44210, 44310, 44410, 44510, 44610, 44710, 44810, 44910,\n",
       "       45010, 45110, 45210, 45310, 45410, 45510, 45610, 45710, 45810,\n",
       "       45910, 46010, 46110, 46210, 46310, 46410, 46510, 46610, 46710,\n",
       "       46810, 46910, 47010, 47110, 47210, 47310, 47410, 47510, 47610,\n",
       "       47710, 47810, 47910, 48010, 48110, 48210, 48310, 48410, 48510,\n",
       "       48610, 48710, 48810, 48910, 49010, 49110, 49210, 49310, 49410,\n",
       "       49510, 49610, 49710, 49810, 49910, 50010, 50110, 50210, 50310,\n",
       "       50410, 50510, 50610, 50710, 50810, 50910, 51010, 51110, 51210,\n",
       "       51310, 51410, 51510, 51610, 51710, 51810, 51910, 52010, 52110,\n",
       "       52210, 52310, 52410, 52510, 52610, 52710, 52810, 52910, 53010,\n",
       "       53110, 53210, 53310, 53410, 53510, 53610, 53710, 53810, 53910,\n",
       "       54010, 54110, 54210, 54310, 54410, 54510, 54610, 54710, 54810,\n",
       "       54910, 55010, 55110, 55210, 55310, 55410, 55510, 55610, 55710,\n",
       "       55810, 55910, 56010, 56110, 56210, 56310, 56410, 56510, 56610,\n",
       "       56710, 56810, 56910, 57010, 57110, 57210, 57310, 57410, 57510,\n",
       "       57610, 57710, 57810, 57910, 58010, 58110, 58210, 58310, 58410,\n",
       "       58510, 58610, 58710, 58810, 58910, 59010, 59110, 59210, 59310,\n",
       "       59410, 59510, 59610, 59710, 59810, 59910, 60010, 60110, 60210,\n",
       "       60310, 60410, 60510, 60610, 60710, 60810, 60910, 61010, 61110,\n",
       "       61210, 61310, 61410, 61510, 61610, 61710, 61810, 61910, 62010,\n",
       "       62110, 62210, 62310, 62410, 62510, 62610, 62710, 62810, 62910,\n",
       "       63010, 63110, 63210, 63310, 63410, 63510, 63610, 63710, 63810,\n",
       "       63910, 64010, 64110, 64210, 64310, 64410, 64510, 64610, 64710,\n",
       "       64810, 64910, 65010, 65110, 65210, 65310, 65410, 65510, 65610,\n",
       "       65710, 65810, 65910, 66010, 66110, 66210, 66310, 66410, 66510,\n",
       "       66610, 66710, 66810, 66910, 67010, 67110, 67210, 67310, 67410,\n",
       "       67510, 67610, 67710, 67810, 67910, 68010, 68110, 68210, 68310,\n",
       "       68410, 68510, 68610, 68710, 68810, 68910, 69010, 69110, 69210,\n",
       "       69310, 69410, 69510, 69610, 69710, 69810, 69910, 70010, 70110,\n",
       "       70210, 70310, 70410, 70510, 70610, 70710, 70810, 70910, 71010,\n",
       "       71110, 71210, 71310, 71410, 71510, 71610, 71710, 71810, 71910,\n",
       "       72010, 72110, 72210, 72310, 72410, 72510, 72610, 72710, 72810,\n",
       "       72910, 73010, 73110, 73210, 73310, 73410, 73510, 73610, 73710,\n",
       "       73810, 73910, 74010, 74110, 74210, 74310, 74410, 74510, 74610,\n",
       "       74710, 74810, 74910, 75010, 75110, 75210, 75310, 75410, 75510,\n",
       "       75610, 75710, 75810, 75910, 76010, 76110, 76210, 76310, 76410,\n",
       "       76510, 76610, 76710, 76810, 76910, 77010, 77110, 77210, 77310,\n",
       "       77410, 77510, 77610, 77710, 77810, 77910, 78010, 78110, 78210,\n",
       "       78310, 78410, 78510, 78610, 78710, 78810, 78910, 79010, 79110,\n",
       "       79210, 79310, 79410, 79510, 79610, 79710, 79810, 79910, 80010,\n",
       "       80110, 80210, 80310, 80410, 80510, 80610, 80710, 80810, 80910,\n",
       "       81010, 81110, 81210, 81310, 81410, 81510, 81610, 81710, 81810,\n",
       "       81910, 82010, 82110, 82210, 82310, 82410, 82510, 82610, 82710,\n",
       "       82810, 82910, 83010, 83110, 83210, 83310, 83410, 83510, 83610,\n",
       "       83710, 83810, 83910, 84010, 84110, 84210, 84310, 84410, 84510,\n",
       "       84610, 84710, 84810, 84910, 85010, 85110, 85210, 85310, 85410,\n",
       "       85510, 85610, 85710, 85810, 85910, 86010, 86110, 86210, 86310,\n",
       "       86410, 86510, 86610, 86710, 86810, 86910, 87010, 87110, 87210,\n",
       "       87310, 87410, 87510, 87610, 87710, 87810, 87910, 88010, 88110,\n",
       "       88210, 88310, 88410, 88510, 88610, 88710, 88810, 88910, 89010,\n",
       "       89110, 89210, 89310, 89410, 89510, 89610, 89710, 89810, 89910,\n",
       "       90010, 90110, 90210, 90310, 90410, 90510, 90610, 90710, 90810,\n",
       "       90910, 91010, 91110, 91210, 91310, 91410, 91510, 91610, 91710,\n",
       "       91810, 91910, 92010, 92110, 92210, 92310, 92410, 92510, 92610,\n",
       "       92710, 92810, 92910, 93010, 93110, 93210, 93310, 93410, 93510,\n",
       "       93610, 93710, 93810, 93910, 94010, 94110, 94210, 94310, 94410,\n",
       "       94510, 94610, 94710, 94810, 94910, 95010, 95110, 95210, 95310,\n",
       "       95410, 95510, 95610, 95710, 95810, 95910, 96010, 96110, 96210,\n",
       "       96310, 96410, 96510, 96610, 96710, 96810, 96910, 97010, 97110,\n",
       "       97210, 97310, 97410, 97510, 97610, 97710, 97810, 97910, 98010,\n",
       "       98110, 98210, 98310, 98410, 98510, 98610, 98710, 98810, 98910,\n",
       "       99010, 99110, 99210, 99310, 99410, 99510, 99610, 99710, 99810,\n",
       "       99910])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var_coin_tosses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "#coin_tosses = np.random.binomial(n=1, p=.5, size=var_coin_tosses)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "#coin_tosses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10,)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coin_tosses.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "number_of_tails = []\n",
    "#prob = []\n",
    "\n",
    "for vct in var_coin_tosses:\n",
    "    \n",
    "    count_tosses = np.random.binomial(n=1, p=.5, size=vct)\n",
    "    number_of_tails.append(np.count_nonzero(count_tosses))\n",
    "    #prob.append(number_of_tails/vct)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3,\n",
       " 50,\n",
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       " 47840,\n",
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       " 48026,\n",
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       " 47998,\n",
       " 48190,\n",
       " 48241,\n",
       " 48407,\n",
       " 48468,\n",
       " 48567,\n",
       " 48506,\n",
       " 48672,\n",
       " 48644,\n",
       " 48604,\n",
       " 49048,\n",
       " 48511,\n",
       " 48648,\n",
       " 48951,\n",
       " 48861,\n",
       " 49067,\n",
       " 49012,\n",
       " 49181,\n",
       " 48950,\n",
       " 49028,\n",
       " 49157,\n",
       " 49391,\n",
       " 49417,\n",
       " 49347,\n",
       " 49504,\n",
       " 49540,\n",
       " 49376,\n",
       " 49679,\n",
       " 49347,\n",
       " 49565,\n",
       " 49527,\n",
       " 49687,\n",
       " 49939,\n",
       " 50123,\n",
       " 49769,\n",
       " 49893]"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "number_of_tails"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "#prob"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "nt = np.array(number_of_tails)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "relative_frequency = nt/var_coin_tosses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.3       , 0.45454545, 0.54285714, 0.51935484, 0.50243902,\n",
       "       0.52745098, 0.47868852, 0.51549296, 0.48148148, 0.49010989,\n",
       "       0.51287129, 0.52342342, 0.52231405, 0.49007634, 0.51985816,\n",
       "       0.51523179, 0.51242236, 0.48070175, 0.48729282, 0.51256545,\n",
       "       0.48557214, 0.51042654, 0.5040724 , 0.49350649, 0.50456432,\n",
       "       0.49482072, 0.49195402, 0.49778598, 0.49466192, 0.50343643,\n",
       "       0.51129568, 0.50771704, 0.51433022, 0.50030211, 0.50733138,\n",
       "       0.4994302 , 0.50277008, 0.50431267, 0.49186352, 0.49232737,\n",
       "       0.48329177, 0.49537713, 0.49952494, 0.50348028, 0.49614512,\n",
       "       0.50088692, 0.50455531, 0.50785563, 0.49334719, 0.49226069,\n",
       "       0.50199601, 0.50234834, 0.49980806, 0.49227872, 0.50277264,\n",
       "       0.50399274, 0.48663102, 0.51330998, 0.47125645, 0.49593909,\n",
       "       0.49517471, 0.50294599, 0.51465378, 0.50427892, 0.50265211,\n",
       "       0.5030722 , 0.49863843, 0.49895678, 0.4958884 , 0.50144718,\n",
       "       0.49814551, 0.50351617, 0.49722607, 0.50793434, 0.49244265,\n",
       "       0.50439414, 0.5021025 , 0.5001297 , 0.49731114, 0.50202276,\n",
       "       0.50873908, 0.49617756, 0.50255786, 0.50036101, 0.49393579,\n",
       "       0.50587544, 0.50708479, 0.50229621, 0.49568672, 0.49506173,\n",
       "       0.50943396, 0.49813392, 0.50130293, 0.49935553, 0.50786397,\n",
       "       0.4873817 , 0.50072841, 0.50267765, 0.49704383, 0.49889001,\n",
       "       0.49400599, 0.50375865, 0.50342801, 0.50514064, 0.49510086,\n",
       "       0.49248335, 0.49915174, 0.4977591 , 0.49130435, 0.50064161,\n",
       "       0.49709355, 0.49783978, 0.50026762, 0.49699381, 0.5035057 ,\n",
       "       0.50130321, 0.50757967, 0.50418446, 0.50211685, 0.49353484,\n",
       "       0.49567027, 0.5       , 0.5003276 , 0.5037368 , 0.4914585 ,\n",
       "       0.4980016 , 0.49730373, 0.49622345, 0.49836066, 0.49178931,\n",
       "       0.49192929, 0.50251716, 0.5000757 , 0.49699474, 0.50029828,\n",
       "       0.49555885, 0.50168993, 0.49773888, 0.50593773, 0.50848311,\n",
       "       0.50463954, 0.49992913, 0.49894441, 0.49468903, 0.49743234,\n",
       "       0.49379738, 0.49247091, 0.5       , 0.49500338, 0.48752515,\n",
       "       0.49666889, 0.49774983, 0.50328731, 0.49954278, 0.49889682,\n",
       "       0.5032882 , 0.50115311, 0.50350095, 0.49569892, 0.49786298,\n",
       "       0.50674578, 0.49422719, 0.50555213, 0.49803801, 0.49792809,\n",
       "       0.50399758, 0.50234798, 0.50341113, 0.50404521, 0.49923122,\n",
       "       0.49853028, 0.49842198, 0.4989541 , 0.50092432, 0.50626077,\n",
       "       0.50022844, 0.50124929, 0.50378317, 0.501516  , 0.49603573,\n",
       "       0.49733481, 0.49751518, 0.49648545, 0.50496996, 0.50597501,\n",
       "       0.50016207, 0.49344438, 0.50213789, 0.49627858, 0.499101  ,\n",
       "       0.50278801, 0.49581371, 0.50468506, 0.49813568, 0.50535806,\n",
       "       0.49764223, 0.49612443, 0.50076104, 0.50020192, 0.50642893,\n",
       "       0.49405297, 0.49975137, 0.49356754, 0.4990645 , 0.50122489,\n",
       "       0.50053632, 0.50053372, 0.499324  , 0.49899087, 0.49827834,\n",
       "       0.50028558, 0.50137376, 0.50075436, 0.50004693, 0.50051378,\n",
       "       0.4976755 , 0.49680703, 0.50446799, 0.5002751 , 0.50314925,\n",
       "       0.4998637 , 0.5027137 , 0.49621792, 0.50228597, 0.50066934,\n",
       "       0.49831186, 0.50238832, 0.49885513, 0.50508549, 0.49349629,\n",
       "       0.49765319, 0.50038944, 0.5       , 0.50523381, 0.5005126 ,\n",
       "       0.50336027, 0.49491741, 0.50269928, 0.50058799, 0.50012547,\n",
       "       0.49995835, 0.49792617, 0.50433705, 0.49954751, 0.49827939,\n",
       "       0.49844961, 0.49475823, 0.50121408, 0.49725917, 0.50140506,\n",
       "       0.50527789, 0.50358423, 0.50329234, 0.50110628, 0.49846517,\n",
       "       0.500392  , 0.50234283, 0.4974718 , 0.50244091, 0.50007719,\n",
       "       0.49946175, 0.49835312, 0.50873712, 0.49661726, 0.49708444,\n",
       "       0.50222558, 0.49748215, 0.49794085, 0.50145468, 0.50174656,\n",
       "       0.50433173, 0.50011066, 0.5031606 , 0.50274625, 0.49868661,\n",
       "       0.49549255, 0.49543644, 0.50057741, 0.49892125, 0.50125403,\n",
       "       0.50039272, 0.50145856, 0.50113435, 0.49805722, 0.50281591,\n",
       "       0.49940372, 0.50206222, 0.4995472 , 0.50159667, 0.50204082,\n",
       "       0.49882799, 0.50395053, 0.50729202, 0.49764585, 0.50442027,\n",
       "       0.50108438, 0.49625127, 0.49764389, 0.5004361 , 0.49478435,\n",
       "       0.50656448, 0.49996679, 0.50069513, 0.49663477, 0.49861888,\n",
       "       0.50134382, 0.49614505, 0.5051449 , 0.50009737, 0.50119702,\n",
       "       0.49851661, 0.50016072, 0.49397629, 0.50827212, 0.51088825,\n",
       "       0.49888924, 0.50028472, 0.5034374 , 0.50546998, 0.50137888,\n",
       "       0.50431115, 0.50283401, 0.4967091 , 0.49774064, 0.50361   ,\n",
       "       0.49569363, 0.49868139, 0.49336594, 0.50252972, 0.501732  ,\n",
       "       0.49784914, 0.49728179, 0.49644685, 0.50075053, 0.50754265,\n",
       "       0.49865712, 0.50130913, 0.49700386, 0.49772257, 0.50224123,\n",
       "       0.49917671, 0.49932571, 0.49871383, 0.50373069, 0.50043592,\n",
       "       0.49965227, 0.50338053, 0.49939499, 0.50540075, 0.50252077,\n",
       "       0.49557269, 0.5009399 , 0.4998864 , 0.49960351, 0.50401017,\n",
       "       0.49994368, 0.50053356, 0.49966396, 0.49491762, 0.50367586,\n",
       "       0.49672313, 0.50354472, 0.49919912, 0.49878821, 0.5021148 ,\n",
       "       0.50566968, 0.50057361, 0.50049033, 0.49883184, 0.50235708,\n",
       "       0.4978114 , 0.50175155, 0.50069874, 0.50064326, 0.49874365,\n",
       "       0.50397227, 0.5039883 , 0.49992045, 0.49846601, 0.4981799 ,\n",
       "       0.50557748, 0.49792705, 0.49617901, 0.50096581, 0.49455871,\n",
       "       0.50033757, 0.5003885 , 0.50105916, 0.50159753, 0.49843228,\n",
       "       0.49794924, 0.49716185, 0.50267789, 0.49661664, 0.50106572,\n",
       "       0.50220197, 0.50244888, 0.49818685, 0.50030143, 0.50328239,\n",
       "       0.49765059, 0.49990027, 0.49917931, 0.50119077, 0.50014848,\n",
       "       0.50394964, 0.49389313, 0.50147384, 0.49953443, 0.50153997,\n",
       "       0.49841502, 0.50007297, 0.50116477, 0.50283224, 0.49814055,\n",
       "       0.49677186, 0.50146599, 0.49652362, 0.50272662, 0.50026247,\n",
       "       0.50107117, 0.5008549 , 0.49789149, 0.50163082, 0.49945768,\n",
       "       0.50150553, 0.49690214, 0.4981269 , 0.50072413, 0.50037287,\n",
       "       0.50197628, 0.50289956, 0.50155057, 0.49974602, 0.50142824,\n",
       "       0.50062055, 0.49935795, 0.50022878, 0.50155216, 0.49945343,\n",
       "       0.49606908, 0.50278848, 0.49728568, 0.49988716, 0.49993245,\n",
       "       0.49858459, 0.49760143, 0.50044733, 0.50441866, 0.50360721,\n",
       "       0.50173295, 0.49807138, 0.4996461 , 0.5035533 , 0.50270865,\n",
       "       0.49898923, 0.50199518, 0.49844673, 0.50065488, 0.50080592,\n",
       "       0.49897848, 0.49891564, 0.49995672, 0.50129562, 0.49767292,\n",
       "       0.50202107, 0.49802617, 0.4995076 , 0.49820551, 0.49720742,\n",
       "       0.50046799, 0.50065803, 0.49760644, 0.49978863, 0.49917739,\n",
       "       0.50014734, 0.50174333, 0.4986376 , 0.50330475, 0.49945732,\n",
       "       0.49604249, 0.49496986, 0.49539515, 0.50091078, 0.49973146,\n",
       "       0.4976912 , 0.50267435, 0.49622254, 0.50120877, 0.49771008,\n",
       "       0.50106101, 0.50008145, 0.49869945, 0.49920909, 0.50032382,\n",
       "       0.50060594, 0.49709736, 0.50203178, 0.49945794, 0.50342617,\n",
       "       0.50131974, 0.50249451, 0.50181239, 0.50075532, 0.504146  ,\n",
       "       0.49871313, 0.4995653 , 0.50427923, 0.50045267, 0.49862502,\n",
       "       0.49925505, 0.49753473, 0.4995704 , 0.4992594 , 0.49848279,\n",
       "       0.49966997, 0.50199574, 0.49535873, 0.50187223, 0.49751493,\n",
       "       0.50073063, 0.49479946, 0.49814212, 0.49973236, 0.49860714,\n",
       "       0.50257094, 0.50347843, 0.50070195, 0.50251846, 0.4986203 ,\n",
       "       0.49726467, 0.5029373 , 0.49560233, 0.4932658 , 0.49692941,\n",
       "       0.50237339, 0.4979668 , 0.49968349, 0.49925664, 0.49972176,\n",
       "       0.49457508, 0.49861393, 0.50188157, 0.49736697, 0.50060651,\n",
       "       0.50045863, 0.49908442, 0.50027417, 0.49906951, 0.50076489,\n",
       "       0.49910925, 0.49729632, 0.50175693, 0.50160911, 0.50265295,\n",
       "       0.50009007, 0.50167236, 0.50305152, 0.50035836, 0.50118047,\n",
       "       0.50226745, 0.50126537, 0.5027931 , 0.50085242, 0.50235774,\n",
       "       0.50164573, 0.49772125, 0.50199259, 0.49795811, 0.50193288,\n",
       "       0.49900018, 0.50099807, 0.50244712, 0.50104694, 0.49892005,\n",
       "       0.49683533, 0.49906266, 0.49701958, 0.50242173, 0.50096702,\n",
       "       0.50299948, 0.4994149 , 0.50146023, 0.50195507, 0.5030303 ,\n",
       "       0.49921381, 0.49786726, 0.49998297, 0.49879272, 0.5009506 ,\n",
       "       0.49905101, 0.49693791, 0.500152  , 0.50020233, 0.50260899,\n",
       "       0.49968073, 0.49927864, 0.5010216 , 0.49963217, 0.49869805,\n",
       "       0.50208299, 0.50316087, 0.50009965, 0.49890565, 0.49942063,\n",
       "       0.50233019, 0.50245834, 0.49963762, 0.49968755, 0.49658513,\n",
       "       0.50073758, 0.50055637, 0.4995589 , 0.50252814, 0.4979645 ,\n",
       "       0.49746383, 0.50071417, 0.50187976, 0.50095454, 0.50479729,\n",
       "       0.49959684, 0.49877636, 0.50036972, 0.50158883, 0.50131389,\n",
       "       0.49982403, 0.50111803, 0.50049434, 0.50175131, 0.49977746,\n",
       "       0.49934931, 0.50076058, 0.49876602, 0.49897331, 0.49910109,\n",
       "       0.49722878, 0.50073888, 0.49781824, 0.49952985, 0.50280081,\n",
       "       0.5012498 , 0.49943847, 0.5017287 , 0.50091743, 0.49978264,\n",
       "       0.5010851 , 0.50082031, 0.50254984, 0.49915137, 0.49759667,\n",
       "       0.50046147, 0.50213485, 0.50246895, 0.5017149 , 0.50253784,\n",
       "       0.49355824, 0.50048773, 0.50054786, 0.50012156, 0.50036413,\n",
       "       0.50368126, 0.50205718, 0.5004531 , 0.49941185, 0.50191236,\n",
       "       0.49941362, 0.49968473, 0.49901064, 0.49679689, 0.49550142,\n",
       "       0.50305924, 0.50138578, 0.50059515, 0.50098054, 0.49676606,\n",
       "       0.49934824, 0.50029581, 0.49856742, 0.50036868, 0.49960241,\n",
       "       0.49677989, 0.49947144, 0.50142208, 0.49980969, 0.49796813,\n",
       "       0.49953291, 0.49874654, 0.50219764, 0.49928789, 0.49773618,\n",
       "       0.50007245, 0.49827811, 0.49747146, 0.50075025, 0.49870336,\n",
       "       0.49769817, 0.50226979, 0.50362932, 0.49628993, 0.50298956,\n",
       "       0.49891444, 0.50061332, 0.49941604, 0.50307211, 0.50231501,\n",
       "       0.49958871, 0.50155785, 0.49869891, 0.49846067, 0.49905514,\n",
       "       0.50166174, 0.49710308, 0.50195197, 0.50071519, 0.50061616,\n",
       "       0.49602853, 0.50099148, 0.4950495 , 0.50226988, 0.49785843,\n",
       "       0.50093043, 0.50205242, 0.5018557 , 0.49889365, 0.49950283,\n",
       "       0.50086885, 0.49987605, 0.50479989, 0.49902486, 0.49662598,\n",
       "       0.501671  , 0.49643004, 0.50142057, 0.5030146 , 0.49847432,\n",
       "       0.50084342, 0.50040755, 0.49864333, 0.49864517, 0.50079827,\n",
       "       0.49795974, 0.49954122, 0.50037731, 0.50117077, 0.50059132,\n",
       "       0.50590525, 0.50158156, 0.49693481, 0.49895736, 0.49803765,\n",
       "       0.50053326, 0.4976834 , 0.50086425, 0.49868543, 0.50063652,\n",
       "       0.4974308 , 0.5022087 , 0.49845463, 0.49763883, 0.50021078,\n",
       "       0.50114459, 0.497635  , 0.49858286, 0.49903027, 0.49729093,\n",
       "       0.5001307 , 0.49801592, 0.50071699, 0.50084624, 0.50367963,\n",
       "       0.50028568, 0.50167293, 0.50170962, 0.49934032, 0.49953494,\n",
       "       0.50178042, 0.49657261, 0.50070776, 0.49988433, 0.49734309,\n",
       "       0.4986412 , 0.49975675, 0.49868303, 0.49697357, 0.49922204,\n",
       "       0.49759266, 0.49768477, 0.50303646, 0.4987565 , 0.50166012,\n",
       "       0.50093659, 0.50011377, 0.50016412, 0.50051696, 0.50215338,\n",
       "       0.50106905, 0.50118076, 0.49416635, 0.50101491, 0.50093856,\n",
       "       0.50052493, 0.49995007, 0.4993517 , 0.50104595, 0.49884343,\n",
       "       0.49802509, 0.50166232, 0.50229216, 0.50071773, 0.49965394,\n",
       "       0.49991359, 0.49780545, 0.50016008, 0.49650719, 0.49986488,\n",
       "       0.49936204, 0.49941184, 0.50061192, 0.49809314, 0.49969479,\n",
       "       0.49879283, 0.50093777, 0.50175161, 0.49968412, 0.50155321,\n",
       "       0.49958793, 0.49938264, 0.4988635 , 0.4978988 , 0.49796165,\n",
       "       0.4992531 , 0.50036097, 0.50122581, 0.50056416, 0.50033569,\n",
       "       0.50053886, 0.49904318, 0.4990921 , 0.49929603, 0.49907043,\n",
       "       0.50188073, 0.49803828, 0.49812374, 0.49997628, 0.50046203,\n",
       "       0.50061531, 0.50104007, 0.50018888, 0.49589671, 0.49969379,\n",
       "       0.50138807, 0.49766185, 0.50160779, 0.50132458, 0.49913359,\n",
       "       0.4967957 , 0.50265156, 0.50078171, 0.50022142, 0.50175765,\n",
       "       0.50139519, 0.50116131, 0.5001276 , 0.49881821, 0.49956024,\n",
       "       0.49890186, 0.49846438, 0.49927344, 0.50013823, 0.50231274,\n",
       "       0.49816113, 0.49950637, 0.50339411, 0.49846524, 0.49652214,\n",
       "       0.50061707, 0.50123274, 0.49936153, 0.5003872 , 0.50095552,\n",
       "       0.49922736, 0.5011236 , 0.5010543 , 0.49697656, 0.50044113,\n",
       "       0.50094905, 0.49814919, 0.49775674, 0.49773674, 0.5014734 ,\n",
       "       0.49689922, 0.49996633, 0.50212981, 0.49920502, 0.5007158 ,\n",
       "       0.49764272, 0.49988841, 0.50163861, 0.50273912, 0.49870982,\n",
       "       0.50047772, 0.49932305, 0.50119721, 0.49900343, 0.50102865,\n",
       "       0.50018782, 0.49952544, 0.5009591 , 0.50143156, 0.50218898,\n",
       "       0.50195583, 0.49838657, 0.49859665, 0.49948527, 0.49974839,\n",
       "       0.49910392, 0.49935597, 0.50062152, 0.50042479, 0.50418888,\n",
       "       0.50092381, 0.49978287, 0.50090012, 0.49771422, 0.49801969,\n",
       "       0.49957842, 0.49817514, 0.50108942, 0.49861006, 0.50165752,\n",
       "       0.50146221, 0.49760498, 0.49872331, 0.49821027, 0.49978589,\n",
       "       0.50252379, 0.49751095, 0.5007043 , 0.49575738, 0.50053242,\n",
       "       0.49865972, 0.50439911, 0.50149666, 0.50129361, 0.50183243,\n",
       "       0.49938631, 0.50009513, 0.49909196, 0.49823858, 0.49833526,\n",
       "       0.50076834, 0.50156661, 0.50045163, 0.49876193, 0.50223247,\n",
       "       0.50183227, 0.50007321, 0.49984328, 0.49902933, 0.50074028,\n",
       "       0.49770857, 0.5032983 , 0.49717285, 0.49812065, 0.49785292,\n",
       "       0.49932649, 0.49933754, 0.50053769, 0.50065076, 0.50115571,\n",
       "       0.50001031, 0.50120482, 0.50040119, 0.4994759 , 0.5035212 ,\n",
       "       0.49749769, 0.49839156, 0.5009825 , 0.49955015, 0.50114391,\n",
       "       0.50007142, 0.50128427, 0.49842175, 0.49870817, 0.49951224,\n",
       "       0.50138057, 0.50113579, 0.49991895, 0.50100192, 0.50085937,\n",
       "       0.4986971 , 0.50125114, 0.49739946, 0.49909375, 0.49820944,\n",
       "       0.49931665, 0.50134525, 0.50268779, 0.49863741, 0.49937944])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "relative_frequency"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.7/site-packages/ipykernel_launcher.py:1: UserWarning: In Matplotlib 3.3 individual lines on a stem plot will be added as a LineCollection instead of individual lines. This significantly improves the performance of a stem plot. To remove this warning and switch to the new behaviour, set the \"use_line_collection\" keyword argument to True.\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<StemContainer object of 3 artists>"
      ]
     },
     "execution_count": 55,
     "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.stem(var_coin_tosses, relative_frequency)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "2"
   ]
  }
 ],
 "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": "3.7.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}