44-Solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Winning an Unfair Game\n",
    "\n",
    "- A game consists of a sequence of plays\n",
    "- On each play either you or your opponent scores a point\n",
    "    - You with probability $p < 1/2$, he with probability $1-p$\n",
    "- The number of plays is to be even\n",
    "- **To win the game, you must get *more* than half the points**\n",
    "- You get to choose in advance the number of plays\n",
    "    - *How many do you choose?*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_____\n",
    "\n",
    "- Let's first think through a few scenarios\n",
    "\n",
    "**2 games**\n",
    "\n",
    "- If we play only two games, then there are only three possible outcomes\n",
    "    1. I win both games\n",
    "        - $P = p^{2}$\n",
    "    2. I lose both games\n",
    "        - $P = (1-p)^{2}$\n",
    "    3. We both win one game\n",
    "        - $P = 2p(1-p)$\n",
    "        \n",
    "- The only way I can win **more** than half the games is to win both\n",
    "    - Therefore, the probability that I win in this scenario is only $p^{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**4 games**\n",
    "\n",
    "- The possibilities are:\n",
    "    1. I **win all 4 games**\n",
    "        - $P=p^{4}$\n",
    "    2. I **lose all 4 games**\n",
    "        - $P=(1-p)^{4}$\n",
    "    3. I **win 1 game**\n",
    "        - $P = \\binom{4}{1}p(1-p)^{3} = 4p(1-p)^{3}$\n",
    "    4. I **win 2 games**\n",
    "        - $P = \\binom{4}{2}p^{2}(1-p)^{2} = 6p^{2}(1-p)^{2}$\n",
    "    5. I **win 3 games**\n",
    "        - $P = \\binom{4}{3}p^{3}(1-p) = 4p^{3}(1-p)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The ways I can win more than half the games is if I:\n",
    "    - Win all 4\n",
    "    - Win 3 out of 4\n",
    "- Therefore, the overall probability that I win is equal to $p^{4} + 4p^{3}(1-p) = p^{3} \\left (p + 4(1-p) \\right ) = p^{3}\\left (4 - 3p \\right )$\n",
    "    - If $p \\leq 1/3$ then this probability is **less than $p^{2}$**\n",
    "    - If $p > 1/3$ then this probability is **greater than $p^{2}$**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "___\n",
    "\n",
    "**6 games**\n",
    "\n",
    "- We know we'll need to win 4, 5, or 6 games so the sum of the probabilities is:\n",
    "\n",
    "$$\n",
    "P(\\text{Winning | 6 games}) = p^{6} + \\binom{6}{5}p^{5}(1-p) + \\binom{6}{4}p^{4}(1-p)^{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "= p^{6} + 6p^{5}(1-p) + 15p^{4}(1-p)^{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "= p^{4}\\left (p^{2} + 6p(1-p) + 15(1-p)^{2} \\right ) = p^{4} \\left (10p^{2} - 24p + 15 \\right )\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- As we can see, there are 3 cases:\n",
    "    1. If $p < 1/3$, we have the highest probability of winning by playing 2 games\n",
    "    2. If $1/3 < p < 0.4$, we have the highest probability of winning by playing 4 games\n",
    "    3. If $0.4 < p < 0.5$, we have the highest probability of winning by playing 6 games"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Let's do one more\n",
    "\n",
    "**8 games**\n",
    "\n",
    "$$\n",
    "P(\\text{Winning | 8 games}) = p^{8} + \\binom{8}{7}p^{7}(1-p) + \\binom{8}{6}p^{6}(1-p)^{2} + \\binom{8}{5}p^{5}(1-p)^{3}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "= p^{8} + 8p^{7}(1-p) + 28p^{6}(1-p)^{2} + 56p^{5}(1-p)^{3}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "= p^{5} \\left (56 - 35p^{3} + 120p^{2} - 140p \\right )\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_______\n",
    "\n",
    "**General case**\n",
    "\n",
    "- Before we continue onto $\\infty$, we notice that if we're playing $2n$ games, the probability that we win is at least half is equal to:\n",
    "\n",
    "$$\n",
    "P_{2n} = \\sum_{n+1}^{2n}\\binom{2n}{i}p^{i}(1-p)^{2n-i}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- If we play $2n+2$ games, then:\n",
    "\n",
    "$$\n",
    "P_{2n+2} = \\sum_{n+2}^{2n+2}\\binom{2n+2}{i}p^{i}(1-p)^{2n+2-i}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- So now we consider the scenario that we've played $2n$ games, and agree to play an *additional* two games for a total of $2n+2$. There are 3 cases for this:\n",
    "    1. We've already won at least $n+2$ of the $2n$ games, therefore **we can lose the next two games and still win overall**\n",
    "    2. We've won no more than $n-1$ games, so **even if we win the next two games, we still lose**\n",
    "    3. We've won $n$ or $n+1$ of the $2n$ games, so **the overall win is still up for grabs**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Let's consider the case where $2n=4$ so $2n+2=6$\n",
    "    - If we had won at least half of the $2n$ games, we won either 3 or 4 of the 4\n",
    "        - To win half of the $2n+2$ games, we need to win 4, 5, or 6 of them\n",
    "            - Therefore, if we won $n+2=2+2=4$ of the $2n$ games initially, we've already won more than half of the $2n+2$ games\n",
    "                - **Case 1**\n",
    "            - If we won only 3 of the first $2n$ games, we need to win at least one of the two remaining games to win overall\n",
    "                - **Case 3**\n",
    "    - If we had won less than half of the $2n$ games, we won either 0 or 1 of them\n",
    "        - In either case, even if we win both of the next two games, we still lose overall\n",
    "            - **Case 2**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- To summarize:\n",
    "\n",
    "$$\n",
    "P(\\text{Winning at least half of the }2n+2\\text{ games | Already won }x\\text{ games}) = \\left\\{\\begin{matrix}\n",
    "1 & ;n + 2 \\leq x\\\\ p^{2} & ;x=n\\\\ p^{2} + 2p(1-p) & ;x= n+1\\\\ 0 & x < n\\end{matrix}\\right.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The probability of winning at least $n+2$ of the first $2n$ games is equal to:\n",
    "\n",
    "$$\n",
    "P(n+2 \\leq x) = \\sum_{n+2}^{2n}\\binom{2n}{i}p^{i}(1-p)^{2n-i}= P_{2n} - \\binom{2n}{n+1}p^{n+1}(1-p)^{n-1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The probability of winning $n$ games is equal to:\n",
    "\n",
    "$$\n",
    "P(x=n) = \\binom{2n}{n}p^{n}(1-p)^{n}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The probability of winning $n+1$ games is equal to:\n",
    "\n",
    "$$\n",
    "P(x=n+1) = \\binom{2n}{n+1}p^{n+1}(1-p)^{n-1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Combining everything, we get:\n",
    "\n",
    "$$\n",
    "P_{2n+2} =(1) \\left (P_{2n} - \\binom{2n}{n+1}p^{n+1}(1-p)^{n-1} \\right ) + p^{2}\\left (\\binom{2n}{n}p^{n}(1-p)^{n} \\right) + (p^{2} + 2p(1-p))\\left ( \\binom{2n}{n+1}p^{n+1}(1-p)^{n-1}\\right )\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "= P_{2n} - \\binom{2n}{n+1}p^{n+1}(1-p)^{n-1} + \\binom{2n}{n}p^{n+2}(1-p)^{n} + \\binom{2n}{n+1}p^{n+3}(1-p)^{n-1} + 2\\binom{2n}{n+1}p^{n+2}(1-p)^{n}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "= P_{2n} + p^{n+1}(1-p)^{n-1} \\left (-\\binom{2n}{n+1} + p(1-p)\\binom{2n}{n} + p^{2}\\binom{2n}{n+1} + 2p(1-p)\\binom{2n}{n+1}\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The next bit is all algebra, so we'll skip to the inequalities..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Condition 1**\n",
    "\n",
    "$$\n",
    "(n-1)(1-p) \\leq np\n",
    "$$\n",
    "\n",
    "**Condition 2**\n",
    "\n",
    "$$\n",
    "n(1-p) \\geq (n+1)p\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Combining these two, we get:\n",
    "\n",
    "$$\n",
    "\\frac{1}{1-2p} - 1 \\leq 2n \\leq \\frac{1}{1-2p} + 1\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- So now, for any $p$, we can calculate the integer $n$ such that this condition holds"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "plt.rcParams['figure.figsize'] = 10, 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve(p):\n",
    "    lower_bound = 1 / (1 - 2 * p) - 1\n",
    "    n_lower = math.ceil(lower_bound / 2)\n",
    "    \n",
    "    upper_bound = 1 / (1 - 2 * p) + 1\n",
    "    n_upper = math.floor(upper_bound / 2)\n",
    "    \n",
    "    assert n_upper == n_lower\n",
    "    return 2*n_lower"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "solve = np.vectorize(solve)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "array_p = np.arange(0.01, 0.5, 0.01)\n",
    "array_n = solve(array_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(array_p, array_n)\n",
    "plt.xlabel('p')\n",
    "plt.ylabel('n')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_______\n",
    "\n",
    "- Let's simulate games for different $p$ values to try to tie out to the optimal $n$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "N_trials = 10000000\n",
    "df_results = pd.DataFrame(columns=['p', 'optimal n', 'best n according to simulation'])\n",
    "\n",
    "for i, p in enumerate(array_p):\n",
    "    optimal_n = solve(p)\n",
    "    \n",
    "    n = 0\n",
    "    p_optimal = 0\n",
    "    n_optimal = 0\n",
    "    stop = False\n",
    "    while not stop:\n",
    "        n += 2\n",
    "        threshold = n / 2 + 1\n",
    "        array_results = np.random.binomial(n, p, size=N_trials)\n",
    "        p_temp = np.mean(array_results >= threshold)\n",
    "        if p_temp > p_optimal:\n",
    "            p_optimal = p_temp\n",
    "            n_optimal = n\n",
    "        else:\n",
    "            stop= True\n",
    "    df_results.loc[i] = [p, optimal_n, n_optimal]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>p</th>\n",
       "      <th>optimal n</th>\n",
       "      <th>best n according to simulation</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.01</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.02</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.03</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.04</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.05</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.06</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.07</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.08</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.09</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.10</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.11</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.12</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.13</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.14</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.15</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.16</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.17</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.18</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.19</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.20</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.21</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.22</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>0.23</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.24</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>0.25</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>0.26</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>0.27</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>0.28</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>0.29</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>0.30</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>0.31</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>0.32</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>0.33</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>0.34</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>0.35</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>0.36</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>0.37</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>0.38</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>0.39</td>\n",
       "      <td>4</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>0.40</td>\n",
       "      <td>6</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>0.41</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>0.42</td>\n",
       "      <td>6</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>0.43</td>\n",
       "      <td>8</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>0.44</td>\n",
       "      <td>8</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>0.45</td>\n",
       "      <td>10</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>0.46</td>\n",
       "      <td>12</td>\n",
       "      <td>12</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>0.47</td>\n",
       "      <td>16</td>\n",
       "      <td>16</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>0.48</td>\n",
       "      <td>26</td>\n",
       "      <td>26</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>0.49</td>\n",
       "      <td>50</td>\n",
       "      <td>40</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       p optimal n best n according to simulation\n",
       "0   0.01         2                              2\n",
       "1   0.02         2                              2\n",
       "2   0.03         2                              2\n",
       "3   0.04         2                              2\n",
       "4   0.05         2                              2\n",
       "5   0.06         2                              2\n",
       "6   0.07         2                              2\n",
       "7   0.08         2                              2\n",
       "8   0.09         2                              2\n",
       "9   0.10         2                              2\n",
       "10  0.11         2                              2\n",
       "11  0.12         2                              2\n",
       "12  0.13         2                              2\n",
       "13  0.14         2                              2\n",
       "14  0.15         2                              2\n",
       "15  0.16         2                              2\n",
       "16  0.17         2                              2\n",
       "17  0.18         2                              2\n",
       "18  0.19         2                              2\n",
       "19  0.20         2                              2\n",
       "20  0.21         2                              2\n",
       "21  0.22         2                              2\n",
       "22  0.23         2                              2\n",
       "23  0.24         2                              2\n",
       "24  0.25         2                              2\n",
       "25  0.26         2                              2\n",
       "26  0.27         2                              2\n",
       "27  0.28         2                              2\n",
       "28  0.29         2                              2\n",
       "29  0.30         2                              2\n",
       "30  0.31         2                              2\n",
       "31  0.32         2                              2\n",
       "32  0.33         2                              2\n",
       "33  0.34         4                              4\n",
       "34  0.35         4                              4\n",
       "35  0.36         4                              4\n",
       "36  0.37         4                              4\n",
       "37  0.38         4                              4\n",
       "38  0.39         4                              4\n",
       "39  0.40         6                              4\n",
       "40  0.41         6                              6\n",
       "41  0.42         6                              6\n",
       "42  0.43         8                              8\n",
       "43  0.44         8                              8\n",
       "44  0.45        10                             10\n",
       "45  0.46        12                             12\n",
       "46  0.47        16                             16\n",
       "47  0.48        26                             26\n",
       "48  0.49        50                             40"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Pretty close!"
   ]
  }
 ],
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