An interview riddle seen by a mathematician.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# For plotting\n",
    "%matplotlib inline\n",
    "import matplotlib\n",
    "matplotlib.rcParams['figure.figsize'] = (12, 6)\n",
    "\n",
    "# For simulations\n",
    "import random\n",
    "random.seed(42)\n",
    "\n",
    "# For data handling\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "# For flattening lists through itertools.from_iterable.chain()\n",
    "import itertools"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solve the riddle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### To know which trader presses which bulb, we need to know the number of divisors of each number between 1 and 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# From https://stackoverflow.com/questions/171765/what-is-the-best-way-to-get-all-the-divisors-of-a-number\n",
    "# 3rd answer\n",
    "\n",
    "def divisors(n):\n",
    "    # get factors and their counts\n",
    "    factors = {}\n",
    "    nn = n\n",
    "    i = 2\n",
    "    while i*i <= nn:\n",
    "        while nn % i == 0:\n",
    "            if not i in factors:\n",
    "                factors[i] = 0\n",
    "            factors[i] += 1\n",
    "            nn //= i\n",
    "        i += 1\n",
    "    if nn > 1:\n",
    "        factors[nn] = 1\n",
    "\n",
    "    primes = list(factors.keys())\n",
    "\n",
    "    # generates factors from primes[k:] subset\n",
    "    def generate(k):\n",
    "        if k == len(primes):\n",
    "            yield 1\n",
    "        else:\n",
    "            rest = generate(k+1)\n",
    "            prime = primes[k]\n",
    "            for factor in rest:\n",
    "                prime_to_i = 1\n",
    "                # prime_to_i iterates prime**i values, i being all possible exponents\n",
    "                for _ in range(factors[prime] + 1):\n",
    "                    yield factor * prime_to_i\n",
    "                    prime_to_i *= prime\n",
    "\n",
    "    # in python3, `yield from generate(0)` would also work\n",
    "    for factor in generate(0):\n",
    "        yield factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "bulbs_switched_by_traders = {bulb: list(divisors(bulb)) for bulb in range(1,101)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 2, 4]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Which traders press the switch of light bulb number 4. Note that the number of traders is odd: the bulb is turned on.\n",
    "bulbs_switched_by_traders[4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 5, 17, 85]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Which traders press the switch of light bulb number 85. The number of traders is even. The bulb is turned off.\n",
    "bulbs_switched_by_traders[85]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: True,\n",
       " 2: False,\n",
       " 3: False,\n",
       " 4: True,\n",
       " 5: False,\n",
       " 6: False,\n",
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       " 9: True,\n",
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       " 42: False,\n",
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       " 70: False,\n",
       " 71: False,\n",
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       " 73: False,\n",
       " 74: False,\n",
       " 75: False,\n",
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       " 78: False,\n",
       " 79: False,\n",
       " 80: False,\n",
       " 81: True,\n",
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       " 94: False,\n",
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       " 97: False,\n",
       " 98: False,\n",
       " 99: False,\n",
       " 100: True}"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# See which bulbs are turned on, which are turned off\n",
    "bulbs_turned_on = {bulb: len(traders) % 2 == 1 for (bulb, traders) in bulbs_switched_by_traders.items()}\n",
    "# x % 2 == 1 tests whether x is odd. It means \"does x % 2 equal 1?\", where % is the modulo operator\n",
    "bulbs_turned_on"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Finding the actual number of bulbs turned on\n",
    "sum(bulbs_turned_on.values())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bringing randomness into the game"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Now the probability to switch the light is p, where 0 ≤ p ≤ 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Instead of working on which traders presses which bulb, we count the number of traders by bulb who would press its switch\n",
    "count_traders_by_bulb = {bulb: len(traders) for (bulb, traders) in bulbs_switched_by_traders.items()}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# How many traders press the switch of light bulb number 4 with probability p\n",
    "count_traders_by_bulb[4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# How many traders press the switch of light bulb number 85 with probability p\n",
    "count_traders_by_bulb[85]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Execute simulations with several values of p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### The number of traders pressing the switch of a bulb follows a Binomial probability law, since pressing the switch follows a Bernoulli law"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def simulate_riddle(p):\n",
    "    return {bulb: sum(np.random.binomial(count_traders, p, 1)) % 2 == 1 \n",
    "            for (bulb, count_traders) in count_traders_by_bulb.items()}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of bulbs turned on: 10\n"
     ]
    },
    {
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       " 99: False,\n",
       " 100: True}"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check that we find the same results than before when p = 1.\n",
    "simulation = simulate_riddle(1)\n",
    "print(\"Number of bulbs turned on: \"+ str(sum(simulation.values())))\n",
    "simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of bulbs turned on: 0\n"
     ]
    },
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     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check with p = 0 that all bulbs are turned OFF of course\n",
    "simulation = simulate_riddle(0)\n",
    "print(\"Number of bulbs turned on: \"+ str(sum(simulation.values())))\n",
    "simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of bulbs turned on: 46\n"
     ]
    },
    {
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       " 100: False}"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Test with p = 0.5, or whatever you want. For easier use of Jupyter notebook, use Ctrl + Enter to run several simulations\n",
    "simulation = simulate_riddle(0.5)\n",
    "print(\"Number of bulbs turned on: \"+ str(sum(simulation.values())))\n",
    "simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def distribution(p, n_simulations=500):\n",
    "    return pd.DataFrame([sum(simulate_riddle(p).values()) for i in range(n_simulations)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[<matplotlib.axes._subplots.AxesSubplot object at 0x000000001420A400>]], dtype=object)"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x14284048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Get the distribution of the number of turned on light bulbs with p = 0.5\n",
    "distribution(0.5, 1000).hist(bins=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# Get the distribution with p varying from 0 to 1 incrementally. It should take less than 20 seconds\n",
    "distribution_3D = [distribution(p, n_simulations=200) for p in [x/100 for x in range(101)]]\n",
    "df_distribution_3D = pd.concat(distribution_3D, axis=1).T.reset_index(drop=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# Empirically, look at how the number of turned on light bulbs evolve from \n",
    "# The synthax below uses \"chaining\" which is made possible in pandas library\n",
    "df_number_of_turned_on_bulbs = df_distribution_3D.mean(axis=1).reset_index() \\\n",
    "    .rename(columns={0: 'Average number of turned on light bulbs', 'index': 'p'}) \\\n",
    "    .assign(p = [x/100 for x in range(101)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>p</th>\n",
       "      <th>Average number of turned on light bulbs</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.00</td>\n",
       "      <td>0.000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.01</td>\n",
       "      <td>4.675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.02</td>\n",
       "      <td>8.795</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.03</td>\n",
       "      <td>12.020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.04</td>\n",
       "      <td>15.790</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.05</td>\n",
       "      <td>18.435</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.06</td>\n",
       "      <td>21.425</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.07</td>\n",
       "      <td>23.545</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.08</td>\n",
       "      <td>26.110</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.09</td>\n",
       "      <td>28.015</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.10</td>\n",
       "      <td>30.565</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.11</td>\n",
       "      <td>32.095</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.12</td>\n",
       "      <td>33.650</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.13</td>\n",
       "      <td>34.845</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.14</td>\n",
       "      <td>36.285</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.15</td>\n",
       "      <td>37.105</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.16</td>\n",
       "      <td>38.175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.17</td>\n",
       "      <td>39.390</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.18</td>\n",
       "      <td>40.105</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.19</td>\n",
       "      <td>40.910</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.20</td>\n",
       "      <td>41.620</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.21</td>\n",
       "      <td>43.180</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>0.22</td>\n",
       "      <td>43.470</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.23</td>\n",
       "      <td>43.975</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>0.24</td>\n",
       "      <td>45.375</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>0.25</td>\n",
       "      <td>45.570</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>0.26</td>\n",
       "      <td>45.655</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>0.27</td>\n",
       "      <td>46.155</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>0.28</td>\n",
       "      <td>46.540</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>0.29</td>\n",
       "      <td>46.525</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>71</th>\n",
       "      <td>0.71</td>\n",
       "      <td>47.110</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>72</th>\n",
       "      <td>0.72</td>\n",
       "      <td>47.805</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>73</th>\n",
       "      <td>0.73</td>\n",
       "      <td>47.300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>0.74</td>\n",
       "      <td>46.815</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75</th>\n",
       "      <td>0.75</td>\n",
       "      <td>45.910</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>76</th>\n",
       "      <td>0.76</td>\n",
       "      <td>45.400</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>77</th>\n",
       "      <td>0.77</td>\n",
       "      <td>45.550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>78</th>\n",
       "      <td>0.78</td>\n",
       "      <td>44.865</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>79</th>\n",
       "      <td>0.79</td>\n",
       "      <td>43.690</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>80</th>\n",
       "      <td>0.80</td>\n",
       "      <td>43.550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>81</th>\n",
       "      <td>0.81</td>\n",
       "      <td>43.800</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>82</th>\n",
       "      <td>0.82</td>\n",
       "      <td>42.100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>83</th>\n",
       "      <td>0.83</td>\n",
       "      <td>41.940</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>84</th>\n",
       "      <td>0.84</td>\n",
       "      <td>41.100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>85</th>\n",
       "      <td>0.85</td>\n",
       "      <td>40.290</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>86</th>\n",
       "      <td>0.86</td>\n",
       "      <td>39.275</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>87</th>\n",
       "      <td>0.87</td>\n",
       "      <td>37.805</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>88</th>\n",
       "      <td>0.88</td>\n",
       "      <td>36.980</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>89</th>\n",
       "      <td>0.89</td>\n",
       "      <td>35.460</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>90</th>\n",
       "      <td>0.90</td>\n",
       "      <td>33.760</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>91</th>\n",
       "      <td>0.91</td>\n",
       "      <td>33.115</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>92</th>\n",
       "      <td>0.92</td>\n",
       "      <td>31.290</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>93</th>\n",
       "      <td>0.93</td>\n",
       "      <td>29.090</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>94</th>\n",
       "      <td>0.94</td>\n",
       "      <td>27.005</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>0.95</td>\n",
       "      <td>25.220</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>0.96</td>\n",
       "      <td>22.315</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>0.97</td>\n",
       "      <td>20.145</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>0.98</td>\n",
       "      <td>17.110</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>0.99</td>\n",
       "      <td>13.510</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>1.00</td>\n",
       "      <td>10.000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>101 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        p  Average number of turned on light bulbs\n",
       "0    0.00                                    0.000\n",
       "1    0.01                                    4.675\n",
       "2    0.02                                    8.795\n",
       "3    0.03                                   12.020\n",
       "4    0.04                                   15.790\n",
       "5    0.05                                   18.435\n",
       "6    0.06                                   21.425\n",
       "7    0.07                                   23.545\n",
       "8    0.08                                   26.110\n",
       "9    0.09                                   28.015\n",
       "10   0.10                                   30.565\n",
       "11   0.11                                   32.095\n",
       "12   0.12                                   33.650\n",
       "13   0.13                                   34.845\n",
       "14   0.14                                   36.285\n",
       "15   0.15                                   37.105\n",
       "16   0.16                                   38.175\n",
       "17   0.17                                   39.390\n",
       "18   0.18                                   40.105\n",
       "19   0.19                                   40.910\n",
       "20   0.20                                   41.620\n",
       "21   0.21                                   43.180\n",
       "22   0.22                                   43.470\n",
       "23   0.23                                   43.975\n",
       "24   0.24                                   45.375\n",
       "25   0.25                                   45.570\n",
       "26   0.26                                   45.655\n",
       "27   0.27                                   46.155\n",
       "28   0.28                                   46.540\n",
       "29   0.29                                   46.525\n",
       "..    ...                                      ...\n",
       "71   0.71                                   47.110\n",
       "72   0.72                                   47.805\n",
       "73   0.73                                   47.300\n",
       "74   0.74                                   46.815\n",
       "75   0.75                                   45.910\n",
       "76   0.76                                   45.400\n",
       "77   0.77                                   45.550\n",
       "78   0.78                                   44.865\n",
       "79   0.79                                   43.690\n",
       "80   0.80                                   43.550\n",
       "81   0.81                                   43.800\n",
       "82   0.82                                   42.100\n",
       "83   0.83                                   41.940\n",
       "84   0.84                                   41.100\n",
       "85   0.85                                   40.290\n",
       "86   0.86                                   39.275\n",
       "87   0.87                                   37.805\n",
       "88   0.88                                   36.980\n",
       "89   0.89                                   35.460\n",
       "90   0.90                                   33.760\n",
       "91   0.91                                   33.115\n",
       "92   0.92                                   31.290\n",
       "93   0.93                                   29.090\n",
       "94   0.94                                   27.005\n",
       "95   0.95                                   25.220\n",
       "96   0.96                                   22.315\n",
       "97   0.97                                   20.145\n",
       "98   0.98                                   17.110\n",
       "99   0.99                                   13.510\n",
       "100  1.00                                   10.000\n",
       "\n",
       "[101 rows x 2 columns]"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Empirically, look at how the number of turned on light bulbs evolve from \n",
    "df_number_of_turned_on_bulbs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x142f2da0>"
      ]
     },
     "execution_count": 182,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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r9qvT5dZPFo6yOhZ66ITKsWmaKyWt7P68RtK83o8ET2Oapn7z7h69uqVEd80f\nrrvmj7A6EgAAHs1ht+lPl46Xw27osZUHNSElUgszE62OhR5ghzwc16MrDujJtfm66bQM3TlvuNVx\nAADwCoZh6L4LxmpCSoR+9MoO5Ve3WB0JPUA5xjE9uSZff/54ny6dkqKfnzeaJdoAADgBAX52/f2a\nKfKzG/rmM1vU2tlldSQcB+UYR/VKVrHuf2e3Fo5N1AMXj5PNRjEGAOBEJUcG6eErJmlfZZN+9kYO\nF+h5OMoxjuiDnEP6yWs7NXd4rB65cqL87PxVAQDgZJ0xIk53zRuhN7aV6rmNRVbHwTHQePAVq/ZV\n6bsvbNfE1Ej949opCvBjVQoAAE7Vd84epjNHxun+t3dre3G91XFwFJRjfMlHu8p1y9IsDY0P1VM3\nTFew/6ms9gcAAP7LZjP08OUTFRcWoG89u0W1LZ1WR8IRUI7xuTe3ler257Zq9KBwvXDrDEUEO6yO\nBACAT4kM9tfj10xRdXOn7nxxm7pcbqsj4X9QjiFJemZDob738nZNT4/Wc7fMUGSwv9WRAADwSeNS\nIvSrJWO1en+1rvzXBpXWt1kdCV9AOYb+vvKAfvFmjuaNitdTN05TaACjFAAA9KUrp6fpz5dP0O6y\nRi16eJXezz5kdSR0oxwPYKZp6oH3c/XHD/ZqycRBeuyaKQp0cPEdAAD94aJJKXrvzrnKiA3R7c9t\n1T2vZ6ut02V1rAGPcjxAud2mfvFWjh7/7KCunpGmP182UQ6WawMAoF8NjgnRK9+crW+cMUQvbCrS\n4r+u0Z5DjVbHGtBoQwNQZWO7rntyk57dUKRvnDFEv7kwkw0+AACwiL+fTfcsGq1nbp6uhjanlvxt\nrZ7bWGh1rAGLcjzArNhToYWPrFZWYa1+f/E43bOILaEBAPAEc4fH6f0752rWkBj97I0crdhTYXWk\nAYlyPEC0O126960c3bw0S4nhgXrnO3N15fQ0q2MBAIAviA0N0D+unaIxSeH6wSs7WMnCApTjAWBf\nRZOW/HWtlq4v1E2nZeiNO2ZrWHyo1bEAAMARBDrs+tvVk9XlMvWd57fKyVrI/Ypy7OOe21ioxX9Z\no5qWDj114zT9cvEYtoMGAMDDZcSG6PcXj9PWonr930d7rY4zoLCgrQ97fWuJfvZGjk4fEacHvz5B\ncWEBVkcCAAA9tHjCIG3Iq9E/PsvTjIxonT0qwepIAwJnjn1UTmmD7nk9WzOHROvJ66dSjAEA8EK/\nOH+MRifSIL6jAAAgAElEQVSF6/sv71AZ88f9gnLsg+paOvXNZ7coKthff71qsvxYvxgAAK8U6LDr\n71dPlrPLre+8sI35435Aa/IxLrep7764TZWNHXrsmsmKDeWMMQAA3iwjNkS/u3icthTW6cGP9lkd\nx+dRjn3MQ8v3avX+av1qyVhNSouyOg4AAOgFSyYm68rpaXr8s4P6NLfS6jg+jXLsQz7IKdffPj2o\nK6ensoYxAAA+5t7FYzQqMUw/enWn6lo6rY7jsyjHPuJAZbN++MoOTUiN1H0XjLU6DgAA6GWBDrse\nvGyC6ls7dd/bu6yO47Moxz6gqd2pbzyTpQA/mx67ejLrGAMA4KPGDorQt88epre2l+nDXeVWx/FJ\nlGMv53S5ddeL21VQ06q/XjVZgyKDrI4EAAD60B1nDdOYpHD97I1s1TJe0esox17M5Tb1g5d3aEVu\npe67YKxmDY2xOhIAAOhjDrtN//f1CapvdereZYxX9DbKsZcyTVM/eyNby3aU6ccLR+ramYOtjgQA\nAPrJmEHh+u684Xp7R5k+yDlkdRyfQjn2QqZp6v53duvFzcX69lnD9K0zh1kdCQAA9LPbzxyqzORw\n/fzNHMYrehHl2As9tHyfnlpboBtPS9cPvjbC6jgAAMAC/x2vaGhz6pdv5Vgdx2dQjr3MYysP6i+f\nHNAV01L1y/PHyDAMqyMBAACLjEoM13fPHq53dh7Se9mMV/QGyrEX+c/6Av3hg1xdMGGQfnvROIox\nAADQN78wXlFQ3WJ1HK9HOfYS7+ws0y/f2qX5oxP04GUTZLdRjAEAwOHxige/PlGdXW597c+r9Pv3\n9qix3Wl1LK9FOfYChTUtuvu1bE1Oi9Rfr5okh53/bAAA4P8bmRimFT84QxdMHKR/rs7TWX9aqWc3\nFKrL5bY6mtehZXm4zi63vvvCNtkM6dErJynQwe53AADgqxLCA/V/X5+gt789R0PjQ/XzN3N07qOr\ntWpfldXRvArl2MP930d7taOkQX+4ZLxSooKtjgMAADxcZnKEXrptph67erLanW5d9+QmfeOZLDk5\ni9wjlGMPtnJvpf65Kk9Xz0jTonFJVscBAABewjAMLRqXpOXfP13fP2eEPtxVoSfX5FsdyytQjj1U\nZVO7fvjKDo1MCNMvzh9jdRwAAOCFAvzs+u684Zo/OkEPf7xfxbWtVkfyeJRjD+R2m/r+SzvU3NGl\nv1zFnDEAADg1v1oyVoYh3btsl0zTtDqOR6Mce6B/rMrTmgPVunfxWI1ICLM6DgAA8HLJkUH6/jkj\n9ElupT7IKbc6jkejHHuYrUV1evCjvTpvXJKumJZqdRwAAOAjbpidrjFJ4brv7V1qYh3ko6Ice5CG\nNqe++8I2JYQH6ncXswMeAADoPX52m3538ThVNnXowY/2WR3HY1GOPYRpmrrn9Z0qb2jXo1dOUkSQ\nw+pIAADAx0xMjdR1Mwdr6foC7SiutzqOR6Ice4hnNxTqvexy/WjBSE0ZHGV1HAAA4KN+sGCk4sMC\n9NM3stlB7wgoxx5gV1mDfv3OHp05Mk63zh1idRwAAODDwgMdunfxWO0qa9TT6wqsjuNxKMcWa+7o\n0ref36aoEIceumyibDbmjAEAQN9alJmos0fF66Hl+1Ra32Z1HI9CObaQaZr66evZKqxp0aNXTFJ0\niL/VkQAAwABgGIZ+dcFYuU1Tv3lnt9VxPArl2EIvZxVr2Y4yfW/+CM0YEmN1HAAAMICkRgfrm2cM\n1fs55couabA6jsegHFtkb3mT7l22S3OGxepbZw2zOg4AABiAbp6Tochghx5cvtfqKB6DcmyB1s4u\n3fH8VoUGOPTQ5RNkZ84YAABYICzQoW+eMVQr91Ypq6DW6jge4bjl2DCMQMMwNhmGscMwjF2GYfyq\n+/5owzCWG4axv/sj64/10K/f2a2DVc165IqJig8LtDoOAAAYwK6bNVixoQH604d7ZZqm1XEs15Mz\nxx2SzjZNc4KkiZIWGoYxU9LdklaYpjlc0oru2ziO4tpWvbS5WDfOztBpw2KtjgMAAAa4YH8/ffus\nodqYX6t1B2usjmO545Zj87Dm7puO7j+mpCWSlnbfv1TShX2S0Mc8tbZANsPQbaeznjEAAPAMV85I\n06CIQM4eq4czx4Zh2A3D2C6pUtJy0zQ3SkowTfNQ90PKJSX0UUaf0dju1Eubi7R4wiAlRjBOAQAA\nPEOAn13fnTdc24vr9UlupdVxLNWjcmyapss0zYmSUiRNNwwj83++burw2eSvMAzjNsMwsgzDyKqq\nqjrlwN7spU3Faul06eY5GVZHAQAA+JJLpqRocEywHvxon9zugXv2+IRWqzBNs17Sp5IWSqowDCNJ\nkro/HvHXDNM0/2ma5lTTNKfGxcWdal6v5XS59dTafM0cEq3M5Air4wAAAHyJw27TXfOHa/ehRr2f\nU251HMv0ZLWKOMMwIrs/D5J0jqRcScskXd/9sOslvdVXIX3B+znlKmto1y1zmDUGAACe6YIJyRoe\nH6qHlu+Va4CePe7JmeMkSZ8ahrFT0mYdnjl+R9IDks4xDGO/pPndt3EEpmnq36vzNCQ2RGePirc6\nDgAAwBHZbYa+f84IHaxq0VvbS62OYwm/4z3ANM2dkiYd4f4aSfP6IpSvySqs086SBv3mwkzZ2PAD\nAAB4sAVjEzV2ULge/ni/Fk8YJId9YO0ZN7C+W4v8a1WeIoMdumRyitVRAAAAjslmM/TDr41UUffe\nDAMN5biPFVS3aPmeCl0zY7CC/O1WxwEAADiuM0fGaVp6lB7+eL9aOrqsjtOvKMd97Km1+XLYbLpu\n9mCrowAAAPSIYRi659zRqm7u0L9X51sdp19RjvtQQ6tTL2eV6IKJgxQfxqYfAADAe0xOi9KizET9\nY9VBVTV1WB2n31CO+9BzmwrV5mTTDwAA4J1+tGCkOrvcemTFPquj9BvKcR/p7HJr6boCzRkWq9FJ\n4VbHAQAAOGFD4kJ11Yw0vbCpWAermq2O0y8ox33k7R1lqmjs0M1zOWsMAAC813fnDVegn01/+mCv\n1VH6BeW4D7R2dunBj/ZqTFK4zhwxcLfMBgAA3i82NEDfOGOoPthVri2FtVbH6XOU4z7w108OqKyh\nXb9aMlaGwaYfAADAu90yN0NxYQH6/Xu5Mk3f3laactzL8qqa9a/Vebp4UrKmpUdbHQcAAOCUBfv7\n6XvzRyirsE4f7a6wOk6fohz3ItM0dd/buxXoZ9fd546yOg4AAECvuWxqiobGhegP7+fK6XJbHafP\nUI570Ue7K7RqX5XuOmcE6xoDAACf4me36e5Fo5VX3eLT20pTjntJW6dL97+9WyMTwnT9LHbDAwAA\nvmf+6PjPt5VuandaHadPUI57yWMrD6i0vk33LxkrPzs/VgAA4HsMw9DPzxujmpYO/elD31zajRbX\nCwprWvT4qjwtmThIM4bEWB0HAACgz0xIjdT1s9L1zIZCbSmsszpOr6Mc94Jfvb1bDpuhn5472uoo\nAAAAfe6HC0YqKTxQ97y+U51dvnVxHuX4FK3YU6FPcit11/wRSgjnIjwAAOD7QgP89JuLMrWvolmP\nf3bQ6ji9inJ8CtqdLv3q7d0aHh+qG05LtzoOAABAvzl7VILOH5+kv35yQAcqm62O02sox6fg1S0l\nKqpt1b2Lx8rBRXgAAGCAuXfxWAX523XP6zvldvvGznk0upPkdpt6cm2+xqdE6LRhXIQHAAAGnriw\nAP3s3NHaXFCnFzYXWR2nV1COT9LKfZXKq2rRzXMyZBiG1XEAAAAs8fWpKZo9NEYPvJerisZ2q+Oc\nMsrxSXpiTb6SIgJ17rgkq6MAAABYxjAM/e6icep0uXXvW7usjnPKKMcnYXdZo9YeqNH1s9OZNQYA\nAANeemyI7pw/XB/sKtcHOeVWxzklNLuT8OTafAU57LpyWprVUQAAADzCrXOHaFRimH733h6vvjiP\ncnyCKpvatWx7mS6bmqKIYIfVcQAAADyCw27T7WcOVVFtq9bn1Vgd56RRjk/Qs+sL5XS7deNpGVZH\nAQAA8CgLxiYqIsihFzcXWx3lpFGOT0C706VnNxZp/ugEpceGWB0HAADAowQ67LpoUrI+zClXXUun\n1XFOCuX4BLyxrVS1LZ26eQ5njQEAAI7k8mmp6nS59ca2UqujnBTKcQ+Zpqkn1uQrMzlcMzKirY4D\nAADgkUYnhWtCaqRe3Fwk0/S+C/Moxz302b4qHahsZtMPAACA47hiWqr2VTRrW3G91VFOGOW4h55Y\nk6+E8ACdN26Q1VEAAAA82uIJgxTsb9dLm7zvwjzKcQ/kljdq9f5qXTcrXf5+/MgAAACOJTTAT4vH\nD9LbO8vU3NFldZwTQtPrgSfXHN704+oZbPoBAADQE5dPT1Vrp0vv7CizOsoJoRwfR3Ftq97YVqpL\np6QoMtjf6jgAAABeYVJqpEYkhOoFL1vzmHJ8HA8t3yebYeiOs4ZZHQUAAMBrGIahy6elaUdxvfYc\narQ6To9Rjo9hd1mj3txeqhtPy1BiRKDVcQAAALzKRZOS5W+36SUvOntMOT6GP36Yq7AAP91+xlCr\nowAAAHid6BB/LchM1BvbStXudFkdp0cox0exIa9GK/dW6Y6zhiki2GF1HAAAAK90xbRUNbQ59eGu\ncquj9Ajl+AhM09QD7+cqKSJQ189OtzoOAACA15o1JEap0UF60UvWPKYcH8GHu8q1vbhe35s/QoEO\nu9VxAAAAvJbNZujyqalan1ejguoWq+McF+X4f3S53Prjh3s1LD5UF09OtjoOAACA1/v61FTZbYb+\ns77Q6ijHRTn+H69uKVFeVYt+tGCk/Oz8eAAAAE5VQnigLp6UrGc3FupQQ5vVcY6J9vcFbZ0u/fnj\nfZqcFqmvjUmwOg4AAIDPuHP+cJmmqUdXHLA6yjFRjr/g6XUFqmjs0N2LRsswDKvjAAAA+IyUqGBd\nPWOwXs4q9ujZY8pxt/rWTj228oDmjYrX9Ixoq+MAAAD4nDvOGiZ/u00PLd9ndZSjohx3++eqPDV1\ndOlHC0daHQUAAMAnxYUF6KY56Vq2o0y7yzxzS2nKsSSny62Xs4p1zugEjUoMtzoOAACAz7pt7lCF\nB/rpoeV7rY5yRJRjSSv3Vqm6uVOXTU21OgoAAIBPiwh26BtnDNXHeyq1pbDO6jhfQTmW9HJWseLC\nAnTmyDirowAAAPi8G09LV2yov/70Ya5M07Q6zpcM+HJc3dyhT3MrdfGkZNY1BgAA6AfB/n769lnD\ntCGvVmsOVFsd50uO2wYNw0g1DONTwzB2G4axyzCMO7vvjzYMY7lhGPu7P0b1fdze9+a2UnW5TX19\naorVUQAAAAaMK2ekKTkySH/6cK9HnT3uyanSLkk/ME1zjKSZku4wDGOMpLslrTBNc7ikFd23vYpp\nmno5q1iT0iI1LD7M6jgAAAADRoCfXXfNH66dJQ36cFeF1XE+d9xybJrmIdM0t3Z/3iRpj6RkSUsk\nLe1+2FJJF/ZVyL6ys6RB+yqa9fUpXIgHAADQ3y6alKyhcSF68KO9crk94+zxCQ3ZGoaRLmmSpI2S\nEkzTPNT9pXJJXrff8stZxQp02HT+hCSrowAAAAw4fnabvnfOCO2vbNYnuZVWx5F0AuXYMIxQSa9J\nuss0zS+t2mweHhQ5Yt03DOM2wzCyDMPIqqqqOqWwvand6dKyHWValJmk8ECH1XEAAAAGpAVjExUb\nGqDXtpRYHUVSD8uxYRgOHS7Gz5mm+Xr33RWGYSR1fz1J0hHrvmma/zRNc6ppmlPj4jxnqbQPd5Wr\nqb1LX5/ChXgAAABWcdhtumjSIK3IrVBtS6fVcXq0WoUh6QlJe0zTfOgLX1om6fruz6+X9Fbvx+s7\nr2SVKCUqSDOHxFgdBQAAYEC7ZEqKnC5Ty7aXWh2lR2eOT5N0raSzDcPY3v3nXEkPSDrHMIz9kuZ3\n3/YKJXWtWnuwWpdOSZHNZlgdBwAAYEAblRiuzORwvbrV+tEKv+M9wDTNNZKO1iDn9W6c/vHallKZ\npnTJZEYqAAAAPMGlk1N039u7tedQo0YnhVuWY8BtCed2m3p1a7FOGxaj1Ohgq+MAAABA0gUTk+Ww\nG5ZfmDfgyvGG/BoV17axtjEAAIAHiQ7x17xRCXpze6mcLrdlOQZcOX41q0RhgX5amJlodRQAAAB8\nwaVTUlTd3KnP9lq3/O+AKsdN7U69l3NIiycMUqDDbnUcAAAAfMEZI+MUG+qvVy0crRhQ5fjN7WVq\nd7pZ2xgAAMADOew2XTgx2dI1jwdMOTZNU0+vzdf4lAhNTI20Og4AAACOwOo1jwdMOV5zoFoHq1p0\nw+x0Hd7XBAAAAJ5mdJK1ax4PmHL89NoCxYb667zxSVZHAQAAwDFcOjlFOaWN2nOosd+PPSDKcUF1\niz7ZW6mrZgxWgB8X4gEAAHgyK9c8HhDl+D/rC2U3DF0zI83qKAAAADgOK9c89vly3NzRpVeyinXe\n+CTFhwdaHQcAAAA9YNWaxz5fjl/fWqKmji7dMDvd6igAAADoIavWPPbpcux2m3p6XYEmpEZqUlqU\n1XEAAADQQw67TRdMSNYnuZVqanf223F9uhyvPlCtvKoW3chZYwAAAK+zMDNRnS63VvbjaIVPl+On\n1+YrLixA545j+TYAAABvM2VwlGJC/PXR7op+O6bPluP86hZ9urdKV89Ik7+fz36bAAAAPstuM3TO\nmAR9mlupji5XvxzTZ1vj0nUFctgNXcXybQAAAF5rwdhENXd0ad3Bmn45nk+W46Z2p17dUqLzxw9S\nfBjLtwEAAHir2cNiFOJv10e7yvvleD5Zjl/dUqJmlm8DAADwegF+dp05Kl7Ld1fI5Tb7/Hg+V45N\n09R/1hdqUlqkJqRGWh0HAAAAp2jB2ERVN3dqa1Fdnx/L58rxrrJG5Ve36MppzBoDAAD4grNGxsnf\nbtOHOX0/WuFz5Xj57goZhnT26HirowAAAKAXhAU6NHtYjD7aXSHT7NvRCp8rxytyKzQ5LUqxoQFW\nRwEAAEAvWTA2UUW1rcotb+rT4/hUOS5vaFdOaaPmcdYYAADAp8wfnSDDkD7s41UrfKocr8g9vHvK\n/NEJFicBAABAb4oLC9DUwVH6cFff7pbnU+X4490VSo0O0vD4UKujAAAAoJctGJuoPYcaVVzb2mfH\n8Jly3NrZpbUHa7pPuRtWxwEAAEAv+9qYREl9O1rhM+V4zf5qdXa5GakAAADwUWkxwRqVGEY57okV\neyoVFuCnaenRVkcBAABAH1kwNlFZhXWqbu7ok9f3iXLsdptakVupM0bGyd/PJ74lAAAAHMGCsYky\nzcPXmvUFn2iSO0rqVd3cwUgFAACAjxudFKbU6KA+G63wiXK8Yk+l7DZDZ46MszoKAAAA+pBhGFow\nJlFrD9Soqd3Z66/vE+X44z0VmjI4SpHB/lZHAQAAQB9bkJmoTpdbK/dW9fpre305Lqk7vI3gOYxU\nAAAADAiT06IUG+qvD/pgtMLry/GKPZWSxJbRAAAAA4TdZuicMYlamVupdqerV1/b68vxx3sqNCQ2\nREPi2BUPAABgoFiUmaiWTpdW76/u1df16nLc3NGljXm1nDUGAAAYYGYNjVFEkEPv5xzq1df16nK8\nel+VOl3sigcAADDQOOw2zR+doI93V6izy91rr+vV5Xj5ngpFBDk0ZXCU1VEAAADQzxZlJqqxvUvr\n82p67TW9thy73KZW7q3SWSPj5Gf32m8DAAAAJ2nO8FiF+Nv1QS+OVnhtq9xWVKfalk7NH8NIBQAA\nwEAU6LDr7NEJ+mhXhVxus1de02vL8cd7KuVnM3T6CHbFAwAAGKgWZSaqpqVTm/Jre+X1vLIcm6ap\n93MOadbQGIUHOqyOAwAAAIucOTJOgQ5br41WeGU53l5cr8KaVi2eMMjqKAAAALBQsL+fzhgRpw92\nlcvdC6MVXlmO39peJn8/mxZmJlodBQAAABZblJmkisYObSuuP+XX8rpy3OVy652dhzRvVDwjFQAA\nANDZo+PlsBu9MlrhdeV4fV6Nqps7tGQiIxUAAACQwgMdmjMsVu/nlMs0T220wuvK8ZvbyhQW6Kcz\nR7JlNAAAAA5blJmkkro27SprPKXX8apy3O506cNd5VqUmahAh93qOAAAAPAQ88ckyG4z9P4pjlZ4\nVTn+JLdSzR1dWjIx2eooAAAA8CDRIf6akRF9yqMVxy3HhmE8aRhGpWEYOV+4L9owjOWGYezv/hh1\n0glOwJvbShUXFqCZQ2L643AAAADwIosyE5VX1aL9lc0n/Ro9OXP8tKSF/3Pf3ZJWmKY5XNKK7tt9\nqqHVqZV7q7R4/CDZbUZfHw4AAABeZsHYRBmG9H52+Um/xnHLsWmaqyT97358SyQt7f58qaQLTzpB\nD32w65A6XW5dOIlVKgAAAPBV8eGBmpIWdUpzxyc7c5xgmuZ/j1ouKeGkE/TQm9vKlBEbonHJEX19\nKAAAAHiphZmJyi1vOunnn/IFeebhieejTj0bhnGbYRhZhmFkVVVVndQxyhvatSG/RhdMGCTDYKQC\nAAAAR7YwM1FxYQEn/fyTLccVhmEkSVL3x8qjPdA0zX+apjnVNM2pcXFxJ3Wwd3aWyTTFxh8AAAA4\nppSoYG28Z95JP/9ky/EySdd3f369pLdOOkEPvLW9TONTIjQkLrQvDwMAAAAfYDuFxRt6spTbC5LW\nSxppGEaJYRg3S3pA0jmGYeyXNL/7dp84WNWs7NIGXTCBs8YAAADoW37He4Bpmlce5Usnf776BLy1\nvUyGIS2mHAMAAKCPefQOeaZpatn2Us0eGqOE8ECr4wAAAMDHeXQ53lHSoIKaVi2ZwHbRAAAA6Hse\nXY4/2lUuu83QgrGJVkcBAADAAODR5Xj1/mpNTotURLDD6igAAAAYADy2HNc0dyinrEGnDz+5tZEB\nAACAE+Wx5XjNgWqZpjR3BOUYAAAA/cNjy/GqfdWKDHZoXHKE1VEAAAAwQHhkOTZNU6v3V2nOsFjZ\nT2GHEwAAAOBEeGQ5zi1vUmVTh05npAIAAAD9yCPL8er9VZKkucNjLU4CAACAgcQjy/GqfdUakRCq\npIggq6MAAABgAPG4ctzW6dKmglqWcAMAAEC/87hyvDG/Rp1dbpZwAwAAQL/zuHK8al+1AvxsmpER\nbXUUAAAADDCeV473V2l6RrQCHXarowAAAGCA8ahyXFbfpgOVzcwbAwAAwBIeVY7/u4Qb6xsDAADA\nCh5Vjlftq1ZCeIBGJIRaHQUAAAADkMeUY5fb1JoD1Zo7PE6GwZbRAAAA6H8eU453ltSroc3JSAUA\nAAAs4zHlePX+ahmGNGcYW0YDAADAGh5Tjlftq9K45AhFh/hbHQUAAAADlEeU48Z2p7YV17OEGwAA\nACzlEeV43YEaudwm88YAAACwlEeU41X7qxQa4KdJaZFWRwEAAMAAZnk5Nk1Tq/ZVadbQGDnslscB\nAADAAGZ5Gy2qbVVJXZtOH84qFQAAALCW5eV4Y16tJGnW0BiLkwAAAGCgs74c59cqOsRfQ+PYMhoA\nAADWsrwcbyqo0fT0aLaMBgAAgOUsLcdl9W0qrm3TjCHRVsYAAAAAJFlcjjflH543np5BOQYAAID1\nLC3HG/NrFRbop1GJ4VbGAAAAACRZfua4RtPSo2W3MW8MAAAA61lWjquaOnSwqkUzGKkAAACAh7Cs\nHG8uYN4YAAAAnsWycrwpv1bB/nZlJkdYFQEAAAD4EsvK8Ya8Gk0ZHCWH3fKllgEAAABJFpXj+tZO\n7a1o0vR0RioAAADgOSwpx1kFdTJN5o0BAADgWSwpxxvza+TvZ9OE1EgrDg8AAAAckSXleFN+rSam\nRirQYbfi8AAAAMAR9Xs5bu7oUk5ZI+sbAwAAwOP0ezneWlgnl9tk3hgAAAAep9/L8cb8GvnZDE0Z\nHNXfhwYAAACOqd/L8ab8WmUmRyjY36+/Dw0AAAAcU782VNOUdhQ36MbT0j+/z+l0qqSkRO3t7f0Z\nBcApCgwMVEpKihwOh9VRAADoNf1ajls7u9TpcmvGkP8/b1xSUqKwsDClp6fLMIz+jAPgJJmmqZqa\nGpWUlCgjI8PqOAAA9Jp+Hato6XDJMKQpg/9/OW5vb1dMTAzFGPAihmEoJiaGd3wAAD6nf8txZ5dG\nJ4YrIujLb8NSjAHvw/9uAQC+6JTKsWEYCw3D2GsYxgHDMO4+3uNbO11fGqnwJG+++aYMw1Bubq7V\nUSwV+v/au/fgKOt7j+PvbyIQUUjEdiiCNeEMhNw3hg0cEiAHahIKRrkNUKQDnNZShGHolMK0M1br\nP5zKGR2EQh2CzOnRGI1UaKvisTRACDSAhmswSkVAsQ3IRbCDhvzOH1m2JOSygWR3CZ/XDDN57t/n\neb67++W3v/09d94ZlONMmzaN1NRUnnnmmQbzX3/9dQ4dOhSUGK529OhRkpOT232/M2fOpKSkBIAf\n/OAHrZ7b1es3ju+ll15qcpvS0lLGjRvXprieeOIJli1bBkBOTg67d+9u0/YiIiKd1XUXx2YWCawE\nxgCJwDQzS2xpmzrnwvbhH0VFRWRnZ1NUVNQu+6utrW2X/dxMAj3nzz77jF27drFv3z4WLlzYYNn1\nFMc3y7Ves2YNiYktvkSa1VJxLCIiIu3nRlqOM4EPnXN/c859BbwMPNTaRt7Y8CuOL1y4QFlZGYWF\nhbz88sv++VOnTuVPf/qTf/pKq97ly5dZtGgRXq+X1NRUfvvb3wL1LXjDhw+noKDAXwQ9/PDDZGRk\nkJSUxPPPP+/fV2FhIQMHDiQzM5Mf/vCHzJs3D4CamhomTpyI1+vF6/Wyffv2a+Jdt24dEyZMID8/\nnwEDBvCzn/3Mv+zqlt+SkhJmzpzpj/3HP/4xQ4cOpX///pSWljJ79mwSEhL861yxcOFCkpKSGD16\nNDU1NQAcOXKE/Px8MjIyGD58uL+FfebMmcyZM4chQ4Y0iAPq+5PPmjWLlJQU0tPT+ctf/gJAbm4u\nn6kxCvIAAA4eSURBVHzyCR6Ph23btvnXLy8vZ+PGjSxatAiPx8ORI0catGqeOnWK2NhY/zUoKChg\n1KhRjB49mtLSUnJycpg0aRKDBg1i+vTpOOcA2LNnDyNHjiQjI4O8vDxOnjzpn5+WlkZaWhorV668\n5jpD/Q/PFi1aRHJyMikpKRQXF/vvdXPHa87V59Lc/QfYunUrw4YNo3///v5W5CVLlrBt2zY8Hs81\nre0A58+fZ+zYscTHxzNnzhzq6uqA5vOhsd/97nd4PB6Sk5OpqKgAYMuWLXg8HjweD+np6XzxxRct\nnp+IiEhncCOjVfQFjl81fQIY0tIG3W6L4O47uzW7/Mk/HOTQp+dvIKRrJd7Tk18+mNTiOhs2bCA/\nP5+BAwdy9913s2fPHjIyMpgyZQqvvPIKY8eO5auvvuLPf/4zq1atorCwkOjoaHbt2sWlS5fIysoi\nNzcXgHfffZcDBw74f8G/du1aevXqxT//+U+8Xi8TJ07k0qVLPPXUU7z77rv06NGDUaNGkZaWBsCC\nBQtYuHAh2dnZHDt2jLy8PKqqqq6JubKykvfee49u3boRHx/P/Pnzuffee1s8zzNnzrBjxw42btxI\nQUEB27dvZ82aNXi9XiorK/F4PFy8eJHBgwfzzDPP8Ktf/Yonn3ySFStW8Oijj7J69WoGDBjAX//6\nV+bOncvmzZuB+hFHysvLiYyMbHC8lStXYmbs37+fw4cPk5ubS3V1NRs3bmTcuHFUVlY2WH/YsGEU\nFBQwbtw4Jk2a1OK5XLnW+/bto1evXpSWlvLee+9x8OBB7rnnHrKysti+fTtDhgxh/vz5bNiwgW9+\n85sUFxfzi1/8grVr1zJr1ixWrFjBiBEjWLRoUZPHWL9+PZWVlezdu5dTp07h9XoZMWIEQJPHy87O\nbjXuTz/9tNn7D3Dy5EnKyso4fPgwBQUFTJo0iaVLl7Js2TL++Mc/NrnPiooKDh06xH333Ud+fj7r\n168P6Bpe8eWXX1JZWcnWrVuZPXs2Bw4cYNmyZaxcuZKsrCwuXLhAVFRUwPsTERG5WXX4UG5m9ijw\nKECvvuE55FNRURELFiwA6luLi4qKyMjIYMyYMSxYsIBLly7x1ltvMWLECG6//Xbefvtt9u3b52/V\nO3fuHB988AFdu3YlMzOzwdBWy5cv5/e//z0Ax48f54MPPuCzzz5j5MiR9OpV34o+efJkqqurAXjn\nnXcadCs4f/48Fy5cuKYv8OjRo4mOjgYgMTGRjz/+uNXi+MEHH8TMSElJoXfv3qSkpACQlJTE0aNH\n8Xg8REREMGXKFAAeeeQRJkyYwIULFygvL2fy5Mn+fV26dMn/9+TJk68pjAHKysqYP38+AIMGDeK+\n++6jurqanj17thhnoB544AH/NQTIzMykX79+AHg8Ho4ePUpMTAwHDhzggQceAODy5cv06dOHs2fP\ncvbsWX+hO2PGDN58880mz2HatGlERkbSu3dvRo4cya5du+jZs2eTxwukOK6oqGj2/kP9tw0REREk\nJiby97//PaBrkZmZSf/+/YH6/txlZWVtKo6nTZsGwIgRIzh//jxnz54lKyuLn/zkJ0yfPp0JEyb4\nz1VERKQzu5Hi+BPg6mqsn29eA86554HnAQYPHtzi986ttfB2hM8//5zNmzezf/9+zIzLly9jZjz9\n9NNERUWRk5PDpk2bKC4uZurUqUD9V+3PPfcceXl5DfZVWlrKHXfc0WD6nXfeYceOHXTv3p2cnJxW\nh76qq6tj586drbbSdev2rxb4yMhIf7/bq0cQaHysK9tEREQ02D4iIqLZfrtmRl1dHTExMde09F5x\n9Tm3t9tuu83fRaDx+TQ+blPXxDlHUlISO3bsaLDu2bNnbzi25u5Be+63ta4aVzQeOeLKdEv50Nr2\nS5YsYezYsbzxxhtkZWWxadMmBg0aFFA8IiIiN6sb6XO8CxhgZnFm1hWYCmxsn7CCp6SkhBkzZvDx\nxx9z9OhRjh8/TlxcnL8v7JQpU3jhhRfYtm0b+fn5AOTl5bFq1Sq+/vprAKqrq7l48eI1+z537hx3\n3XUX3bt35/Dhw+zcuRMAr9fLli1bOHPmDLW1tbz22mv+bXJzc3nuuef8080VpM3p3bs3VVVV1NXV\n+Vus26Kurs7fIv7SSy+RnZ1Nz549iYuL49VXXwXqC7a9e/e2uq/hw4fz4osvAvXX6NixY8THx7e4\nTY8ePRr0bY2NjWXPnj0ATY7i0Jr4+Hhqamr8xfHXX3/NwYMHiYmJISYmhrKyMgB/nE2dQ3FxMZcv\nX6ampoatW7eSmZnZ5jiu1tL9b07j69JYRUUFH330EXV1dRQXF/tbsAPNhyt9qcvKyoiOjiY6Opoj\nR46QkpLC4sWL8Xq9t/xILiIicmu47uLYOVcLzAM2AVXAK865g+0VWLAUFRUxfvz4BvMmTpzoH7Ui\nNzeXLVu28J3vfIeuXbsC9UNyJSYmcv/995OcnMyPfvSjJlsN8/Pzqa2tJSEhgSVLljB06FAA+vbt\ny89//nMyMzPJysoiNjbW30Vi+fLl7N69m9TUVBITE1m9enWbzmfp0qWMGzeOYcOG0adPnzZfjzvu\nuIOKigqSk5PZvHkzjz/+OFBfPBYWFpKWlkZSUhIbNmxodV9z586lrq6OlJQUpkyZwrp16xq0ijZl\n6tSpPP3006Snp3PkyBF++tOfsmrVKtLT0zl16lSbz6dr166UlJSwePFi0tLS8Hg8lJeXA/DCCy/w\n2GOP4fF4mm2hHT9+PKmpqaSlpTFq1Ch+/etf861vfavNcVytpfvfnNTUVCIjI0lLS2vyB3ler5d5\n8+aRkJBAXFycP6cDzYeoqCjS09OZM2cOhYWFADz77LMkJyeTmppKly5dGDNmzA2ctYiIyM3BAv3a\ntj0MHjzYNR5PtaqqioSEhKDFEC6u9COura1l/PjxzJ49+5oiXTqvznL/b9XXr4iIhD8z2+OcG9zW\n7YL6hDz5lyeeeMI/dFZcXBwPP/xwqEOSINL9FxERCU8dPlqFNO3K08nk1qT7LyIiEp7UciwiIiIi\n4hMWxXEw+z2LSPvQ61ZERDqjkBfHUVFRnD59Wh+0IjcR5xynT5/WU/NERKTTCXmf4379+nHixAlq\nampCHYqItEFUVJSemiciIp1OyIvjLl26NHjcsoiIiIhIqIS8W4WIiIiISLhQcSwiIiIi4qPiWERE\nRETEJ6iPjzazL4D3g3ZAuVl8AzgV6iAk7CgvpCnKC2mK8kKaEu+c69HWjYL9g7z3r+cZ19K5mdlu\n5YU0pryQpigvpCnKC2mKme2+nu3UrUJERERExEfFsYiIiIiIT7CL4+eDfDy5OSgvpCnKC2mK8kKa\noryQplxXXgT1B3kiIiIiIuFM3SpERERERHw6pDg2s3wze9/MPjSzJU0sNzNb7lu+z8zu74g4JHwE\nkBPTfbmw38zKzSwtFHFKcLWWF1et5zWzWjObFMz4JDQCyQszyzGzSjM7aGZbgh2jBF8AnyPRZvYH\nM9vry4tZoYhTgsvM1prZP8zsQDPL21xztntxbGaRwEpgDJAITDOzxEarjQEG+P49Cqxq7zgkfASY\nEx8BI51zKcBTqP9YpxdgXlxZ77+At4MboYRCIHlhZjHAb4AC51wSMDnogUpQBfh+8RhwyDmXBuQA\n/21mXYMaqITCOiC/heVtrjk7ouU4E/jQOfc359xXwMvAQ43WeQj4H1dvJxBjZn06IBYJD63mhHOu\n3Dl3xje5E+gX5Bgl+AJ5rwCYD7wG/COYwUnIBJIX3wPWO+eOATjnlBudXyB54YAeZmbAncDnQG1w\nw5Rgc85tpf5eN6fNNWdHFMd9geNXTZ/wzWvrOtJ5tPV+/yfwZodGJOGg1bwws77AePTt0q0kkPeL\ngcBdZlZqZnvM7PtBi05CJZC8WAEkAJ8C+4EFzrm64IQnYazNNWewn5An0iIz+w/qi+PsUMciYeFZ\nYLFzrq6+MUgEqP/sygBGA7cDO8xsp3OuOrRhSYjlAZXAKODfgP8zs23OufOhDUtuNh1RHH8C3HvV\ndD/fvLauI51HQPfbzFKBNcAY59zpIMUmoRNIXgwGXvYVxt8Avmtmtc6514MTooRAIHlxAjjtnLsI\nXDSzrUAaoOK48wokL2YBS139GLUfmtlHwCCgIjghSphqc83ZEd0qdgEDzCzO1xF+KrCx0Tobge/7\nfkE4FDjnnDvZAbFIeGg1J8zs28B6YIZaf24ZreaFcy7OORfrnIsFSoC5Kow7vUA+QzYA2WZ2m5l1\nB4YAVUGOU4IrkLw4Rv23CZhZbyAe+FtQo5Rw1Oaas91bjp1ztWY2D9gERAJrnXMHzWyOb/lq4A3g\nu8CHwJfU/29POqkAc+Jx4G7gN75Wwlrn3OBQxSwdL8C8kFtMIHnhnKsys7eAfUAdsMY51+QwTtI5\nBPh+8RSwzsz2A0Z9l6xTIQtagsLMiqgfneQbZnYC+CXQBa6/5tQT8kREREREfPSEPBERERERHxXH\nIiIiIiI+Ko5FRERERHxUHIuIiIiI+Kg4FhERERHxUXEsIiIiIuKj4lhERERExEfFsYhImDKzWDM7\nbGYvmlmVmZX4nggnIiIdRMWxiEh4iwd+45xLAM4Dc0Mcj4hIp6biWEQkvB13zm33/f2/QHYogxER\n6exUHIuIhDfXyrSIiLQjFcciIuHt22b2776/vweUhTIYEZHOTsWxiEh4ex94zMyqgLuAVSGOR0Sk\nU7st1AGIiEiLap1zj4Q6CBGRW4VajkVEREREfMw5/bZDRERERATUciwiIiIi4qfiWERERETER8Wx\niIiIiIiPimMRERERER8VxyIiIiIiPiqORURERER8/h9bOTCBdTomIgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x146180f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's plot the result\n",
    "df_number_of_turned_on_bulbs.plot(x='p', y='Average number of turned on light bulbs')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Get the results mathematically"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# If n traders should press the switch a light bulb, the probability that an even number of them press the switch is:\n",
    "# 0.5 * [1 + (1 - 2p)^n]\n",
    "def probability_odd_number_traders(n, p):\n",
    "    return 1 - 0.5 * (1 + (1 - 2*p)**n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "def probabilitiy_by_bulb(p):\n",
    "    return {bulb: probability_odd_number_traders(count_traders, p) \n",
    "            for (bulb, count_traders) in count_traders_by_bulb.items()}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1: 0.5,\n",
       " 2: 0.5,\n",
       " 3: 0.5,\n",
       " 4: 0.5,\n",
       " 5: 0.5,\n",
       " 6: 0.5,\n",
       " 7: 0.5,\n",
       " 8: 0.5,\n",
       " 9: 0.5,\n",
       " 10: 0.5,\n",
       " 11: 0.5,\n",
       " 12: 0.5,\n",
       " 13: 0.5,\n",
       " 14: 0.5,\n",
       " 15: 0.5,\n",
       " 16: 0.5,\n",
       " 17: 0.5,\n",
       " 18: 0.5,\n",
       " 19: 0.5,\n",
       " 20: 0.5,\n",
       " 21: 0.5,\n",
       " 22: 0.5,\n",
       " 23: 0.5,\n",
       " 24: 0.5,\n",
       " 25: 0.5,\n",
       " 26: 0.5,\n",
       " 27: 0.5,\n",
       " 28: 0.5,\n",
       " 29: 0.5,\n",
       " 30: 0.5,\n",
       " 31: 0.5,\n",
       " 32: 0.5,\n",
       " 33: 0.5,\n",
       " 34: 0.5,\n",
       " 35: 0.5,\n",
       " 36: 0.5,\n",
       " 37: 0.5,\n",
       " 38: 0.5,\n",
       " 39: 0.5,\n",
       " 40: 0.5,\n",
       " 41: 0.5,\n",
       " 42: 0.5,\n",
       " 43: 0.5,\n",
       " 44: 0.5,\n",
       " 45: 0.5,\n",
       " 46: 0.5,\n",
       " 47: 0.5,\n",
       " 48: 0.5,\n",
       " 49: 0.5,\n",
       " 50: 0.5,\n",
       " 51: 0.5,\n",
       " 52: 0.5,\n",
       " 53: 0.5,\n",
       " 54: 0.5,\n",
       " 55: 0.5,\n",
       " 56: 0.5,\n",
       " 57: 0.5,\n",
       " 58: 0.5,\n",
       " 59: 0.5,\n",
       " 60: 0.5,\n",
       " 61: 0.5,\n",
       " 62: 0.5,\n",
       " 63: 0.5,\n",
       " 64: 0.5,\n",
       " 65: 0.5,\n",
       " 66: 0.5,\n",
       " 67: 0.5,\n",
       " 68: 0.5,\n",
       " 69: 0.5,\n",
       " 70: 0.5,\n",
       " 71: 0.5,\n",
       " 72: 0.5,\n",
       " 73: 0.5,\n",
       " 74: 0.5,\n",
       " 75: 0.5,\n",
       " 76: 0.5,\n",
       " 77: 0.5,\n",
       " 78: 0.5,\n",
       " 79: 0.5,\n",
       " 80: 0.5,\n",
       " 81: 0.5,\n",
       " 82: 0.5,\n",
       " 83: 0.5,\n",
       " 84: 0.5,\n",
       " 85: 0.5,\n",
       " 86: 0.5,\n",
       " 87: 0.5,\n",
       " 88: 0.5,\n",
       " 89: 0.5,\n",
       " 90: 0.5,\n",
       " 91: 0.5,\n",
       " 92: 0.5,\n",
       " 93: 0.5,\n",
       " 94: 0.5,\n",
       " 95: 0.5,\n",
       " 96: 0.5,\n",
       " 97: 0.5,\n",
       " 98: 0.5,\n",
       " 99: 0.5,\n",
       " 100: 0.5}"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check the probability for p = 0.5, or whatever p\n",
    "probabilitiy_by_bulb(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x14cbe9b0>"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Y5VwAAOiTvjV7sBaOS9bv/r1Xy/aU2h3HJ1COvVBtU7vufXGzIoMD9NjXJirQ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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x14cbe208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot = pd.DataFrame({'p': [x/100 for x in range(101)], \n",
    "                     'Average number of turned on light bulbs': [sum(probabilitiy_by_bulb(x/100).values()) for x in range(101)]}) \\\n",
    "    .plot(x='p', y='Average number of turned on light bulbs')\n",
    "plot.axhline(y=50, color='grey')\n",
    "# The maximum number of turned on light bulbs is reached only at p = 0.5!"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
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