23-Solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Players A and B match pennies N times\n",
    "- They keep a tally of their gains and losses\n",
    "- *After the first toss, what is the chance that at no time during the game will they be even?*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "____\n",
    "\n",
    "**Note**: *\"It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), then Even keeps both pennies, so wins one from Odd (+1 for Even, −1 for Odd). If the pennies do not match (one heads and one tails) Odd keeps both pennies, so receives one from Even (−1 for Even, +1 for Odd).\"*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- After the first toss, there are two possibilities\n",
    "    1. A ahead by 1\n",
    "    2. B ahead by 1\n",
    "    \n",
    "- They have $N-1$ tosses remaining"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Let's consider the possible paths when $N=3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\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": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "array_flips = (np.array([-1, 1]),)*3\n",
    "array_permutations = np.meshgrid(*array_flips)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "df_paths = pd.DataFrame([x.ravel() for x in array_permutations]).cumsum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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       "\n",
       "    .dataframe thead th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-1</td>\n",
       "      <td>-1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>-1</td>\n",
       "      <td>-1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-2</td>\n",
       "      <td>-2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-3</td>\n",
       "      <td>-1</td>\n",
       "      <td>-1</td>\n",
       "      <td>1</td>\n",
       "      <td>-1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   0  1  2  3  4  5  6  7\n",
       "0 -1 -1  1  1 -1 -1  1  1\n",
       "1 -2 -2  0  0  0  0  2  2\n",
       "2 -3 -1 -1  1 -1  1  1  3"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- As we can see, the probability of reaching a tie at some point is 1/2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Let's try $N=6$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "array_flips = (np.array([-1, 1]),)*6\n",
    "array_permutations = np.meshgrid(*array_flips)\n",
    "df_paths = pd.DataFrame([x.ravel() for x in array_permutations]).cumsum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(44, 64)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len([x for x in df_paths.columns if (df_paths[x]==0).sum()>=1]),len(df_paths.columns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- For $N=6$, the probability is 11/16\n",
    "    - **As we can see, the probability isn't just 1/2**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Since each coin flip has two possible outcomes, there are $2^{N}$ different paths the game can take"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Let's visualize the simulated probabilities for different $N$ values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sim(N, n_trials):\n",
    "    array_flips = np.random.randint(0, 2, size=(N, 2, n_trials))\n",
    "    array_sum = np.sum(array_flips, axis=1)\n",
    "    array_boolean = (array_sum == 1).astype(int)\n",
    "    array_score = np.cumsum(array_boolean + (-1 + array_boolean), axis=0)\n",
    "    array_results = (np.sum(array_score == 0, axis=0)>0).astype(int)\n",
    "    return np.mean(array_results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_trials = 100000\n",
    "x = []\n",
    "list_results = []\n",
    "\n",
    "for N in range(2, 100):\n",
    "    x.append(N)\n",
    "    list_results.append(sim(N, n_trials))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9HQGMHkEGABo8QqK9p1879x/Qyzs79dz2/Xpld0gDiaQCPqN5NcU6Z06l5tcUa251sebXlKi6JMjVMACOIMgAZKW27qh+9tcd2tHeq8auiBo7+9QfTx78/MJJJfrgWdN01uxKrZhezqGqAFzF/8IAyDobmkK6+ddr1HlgQLMnFmlWVaEumFelKeUFmlJeoCV1ZdzjEcAJRZAByCoPrW/Rl+5br8qioB645WwtmMRBqgC8R5AByArJpNV/PLFVt61s0Irp5fqvG09VRVHQ67EAQBJBBiDDJZJWTV19+n+PbNZjG1t1/WlTdOvVJys34PN6NAA4iCADMG4d6I+ro3dA3dHY4Fskrp5oTM2hiBraetXQ1qud+w+oP56Uz0jfuHKhbjprOq+MBDDmEGQAxoWBeFL1+3q0rimk9Y2Dbw3tvbL2zY81RqqbkK/ZVUU6Z3alZk8s0qnTJmhudfGJHxwAUkCQARjT4omkfvncLv3gia06MJCQJFUU5uqUKWV6++LJqp2Qr+K8gEryclScF1Bpfo4qi4LKz/V7PDkApI4gAzBmbdnXra/cv0Hrm8K6aP5EvfPUOi2ZUspNtQFkHIIMwJjTH0/oxysb9JOnt6s0P0c/eu9SXbFoEhEGIGMRZABOuOZQRC/t6NBLOzq1tzuqXL9PuQEz9KtPr+wJqaGtV+9cWquvv32hJnBIK4AMR5ABcN2+cFTPNezXCzs69NLODjV2RiRJpfk5ml5ZqFg8qYFEUgPxwbeS/IB+edNpumD+RI8nB4ATgyAD4KhE0qonGtOqXV16rmG/nm3Yr4a2XknShIIcrZhRrg+dNUNnzKzQ/Jpi+XwsQwIAQQZg1FZuadWtD21SdzSugXhS/fGEYonXz6HIy/Hp9BkVes/yKTp7diUBBgBHQZABGJXVuzr1yd+8omkVBbp8UY2CAb+CAZ+CAb/yc31aVFumU6eVKRjg+AkAGAlBBiBt9ft69OFfrVJtWb7u/tgZ3BMSAI4TN3MDkJamrj594I6XlJfj150fXkGMAYADuEIGIGWdBwb0gTteVt9AQvd+/ExNKS/weiQAyAgEGYCU9A3E9eFfrVJTV0S/+cjpWjCpxOuRACBjEGRAFrPW6onNbUpaq3nVxZpSXiD/Ia+CDPfF9PTWNj2xuU1P17fpQH9cP71xmVbMKPdwagDIPAQZkKWstfqXP23Wz5/defBjeTk+zZlYrDnVRWoJRbRqV5cSSavKolxddnKN3nlqnc6YWeHh1ACQmQgyIAtZa3Xrw5v0y+d26aazpuuapbWqb+3R1n09qm/t0XMN+zWhIFefOG+m3rqgWkvqyjg/DABcRJABWcZaq396aJN+9fwuffjsGfr62xfIGKMlU8q8Hg0AshZBBmQRa62+8eBG/fqF3froOTP0f68YjDEAgLcIMiADvbC9Q4++tleFwYBK83MOvv11a7vuWdWom8+dqa9dNp8YA4AxgiADMkg8kdR/PLFVP3l6u/ICfsUSScWT9g2P+fh5M/XVS4kxABhLCDIgQzR19elz96zTmt1deteyOv3T1ScpP8evvoGEwpGYwpGYjJHmVRcTYwAwxhBkwDiytbVHrd1R1ZTkqaY0T8V5OZKkP7+2V393/wYlrfSf15+iq0+pPfg1hcGACoMBTS7L92psAMAICDJgnFi1q1PX3/6iEocsQRbm+lVZHNTujj4trivVbTcs1bSKQg+nBACMBkEGjANdBwb02bvXqm5Cvr59zSK19/ZrXziqfd1RtXZHdc3SWn3q/NnKDfi8HhUAMAoEGTDGJZNWX7xvvTp6B/Q/nzpLJ9eWej0SAMBh/N9pwEPxRFLf/fMWveVfV+rPr+074mN+8exOrdzSpv97xQJiDAAyFEEGeKStJ6obf/GS/uvp7YrFrT7xmzX68n3r1dsfP/iYV/Z06bt/3qJLT6rRB86c5uG0AAA3sWQJeGDVrk59+revqDsa07+/a4muXDJZP3xym37ydINe3Nmh77/7FM2dWKzP/G6takrz9N3rFnNUBQBkMIIMOIGstfrFszv17Ue3aMqEfN354RVaMKlEkvSlS+bp/HlV+sK96/Wen72gGZWFauuJ6r5PnKXS/ByPJwcAuIkgA04Aa62e2dqunz6zXS/u6NTbFlbr3969RCV5bwyt5dPL9cjn3qJvPbRJv1/dqK+/faFO4abfAJDxjLV25EeNIcuXL7erV6/2egwgJf3xhB5c16Kf/22n6lt7VF0S1KcvmK33nzFtxCXI9p5+VRUHT9CkAAA3GGPWWGuXj/Q4rpABLrnrxd364ZPb1N7Tr/k1xfr+u5fo7Ysnp3xWGDEGANmDIANccNcLu/T1Bzbq9Bnl+v67l+ic2ZVsygcAHBVBBjjssY379I8PbtRbF0zUT29cpoCf02UAAMfG3xSAg9bs7tRn716rJXVluu2GU4kxAEBK+NsCcMj29l595M7VmlSap198cLnyc/1ejwQAGCdYsgTS0NHbrzuf36W6CQWaU12kOdXFKgoG1NYT1QfveFkBn9GdH16hiiI25AMAUkeQAWn4zqNbdN+apjd8rG5CvpJJq66+mO65+QxNqyj0aDoAwHhFkAEp2rKvW/e/0qQPnz1DHzhzmupbe7SttUf1rb3aG4ro29cu1hIOcQUAjAJBBqToO49uUXEwoM9eNFtlBbmaXlmoS06q8XosAEAGYFM/kILnG/br6fp23XLhYIwBAOAkggwYQTJp9e1Ht6i2LF8fOHO61+MAADIQQQaM4KENLXq1OawvXTJXeTkcZQEAcB5BBhxDfzyh7z1Wr4WTSnT1klqvxwEAZCiCDDiG37y4R01dEX3t8vny+bgXJQDAHQQZcBThSEy3rdymt8yp1FvmVHk9DgAgg3HsBbJeMmn1y+d36aUdHeqOxhSOxNUdiamrb0CRWEJfvWy+1yMCADIcQYas1tsf1xd+v05/2dSqmVWFqiwMqrYsXwsmFas0P0enzyjXSZNLvR4TAJDhCDJkrZ37D+jmX6/Wjv0H9I0rF+qms6bLGPaJAQBOPIIMWemp+jZ99u61CviM7vrICp01q9LrkQAAWYwgQ9ZIJq0a2nv18PoW3fZUg+bXlOj29y/TlPICr0cDAGQ5ggwZbe2eLj2/vUNrdndpze4uhSMxSdKVSybrX69drPxcDnoFAHiPIEPGemh9iz5z91pJ0qyqQl16Uo2WT5+g5dPLNaOy0OPpAAB4HUGGjBSNJfSdR7do4aQS/eajp6u8kBuCAwDGLg6GRUa664Xdag5F9PeXLyDGAABjHkGGjBPqG9BtK7fpvLlVOmcOr54EAIx9BBkyzo9WNqi3P66vXc4J+wCA8YEgQ0Zp7OzTr1/YreuW1Wl+TYnX4wAAkBKCDBnlXx+rl88nfeHieV6PAgBAyggyZIz1jSE9tL5FHz1npmpK87weBwCAlBFkyAjWWv3LI5tVUZirj5830+txAABIC+eQYdz57p+36MF1LcrL8SkY8CsvxydjjNbs7tK3rj5JxXk5Xo8IAEBaCDKMK3s6+nT7X3fo5NpS1ZXlqz+eUDSWVH88oSsWTdL1K6Z6PSIAAGkjyDCu/OTpBvl9Rre/f5mqS9gnBgDIDOwhw7jRHIroD6806T3LpxBjAICMQpBh3PjZM9slSZ84f5bHkwAA4CyCDONCW3dU96xq1LWn1qm2LN/rcQAAcBRBhnHh9r/uUCJp9UmujgEAMhBBhjGvo7dfv31pj65eMlnTKgq9HgcAAMcRZBgTBuJJ/eLZnXqtOfymz/3i2Z2KxhP61AWzPZgMAAD3cewFxoT71jTqWw9vkiSdPbtCN587S+fOqVQ4EtOvX9ityxdN0uyJRR5PCQCAOwgyeC6eSOqnz2zX4rpSXb5okn753E598I6XNb+mWNMqCtTbH9dnLuTqGAAgc7FkCc89vGGvGjsjuuWC2frEebP0t7+7UN+7brGS1uqxja1628Jqza8p8XpMAABcwxUyeCqZtPrxUw2aV12sty6oliTlBnx61/IpuvbUOq3e3aW51SxVAgAyG1fI4KnHN7dqW1uvPnXBLPl85g2f8/mMVswoV1lBrkfTAQBwYhBk8Iy1Vj95qkFTywt0xaJJXo8DAIBnCDJ45tmG/VrfFNYnz5+lgJ8/igCA7MXfgvDMj59qUHVJUO88tdbrUQAA8BRBBk+s2d2pF3d06mNvmalgwO/1OAAAeIogg+viieSbPvbjp7ZrQkGO3nv6VA8mAgBgbOHYC7jq1oc26Y7ndqokL6DKoqAqinJVmp+rlVva9MWL56oglz+CAADwtyFcE40ldN/qRp06tUyLaku1/8CAOnr7tafzgE6aXKIPnDnd6xEBABgTCDK4ZuWWNvX0x/X5i+fqLXOqvB4HAIAxiz1kcM0f1zarqjios2ZVej0KAABjGkEGV4T7Ynq6vl1XLZks/2En8AMAgDciyOCKR17bq4FEUu84hTPGAAAYCUEGV/zv2mbNrCrUybUlXo8CAMCYR5DBcc2hiF7e2alrTqmVMSxXAgAwEoIMjntwXYsk6WqWKwEASAlBBsc9sK5Zp04t09SKAq9HAQBgXCDI4KjNe7u1ZV+P3rGUq2MAAKSKIEParLW6b3WjGtp63/S5P65rlt9ndMWiSR5MBgDA+MRJ/Ujb2saQvnz/BuX4jT5yzkx95sLZKgwGlExaPbSuRefNrVJFUdDrMQEAGDe4Qoa0PbZxnwI+oysXT9ZPn9mut37/Gf1pw169tLNTLeGorj5lstcjAgAwrnCFDGmx1uovG1t15qwKff89p+h9Z0zV1/+4UZ/+3SsqyQuoINevixdWez0mAADjiqtXyIwxlxpj6o0xDcaYrx7h81ONMU8ZY9YaYzYYYy53cx4cv4a2Xu3cf0BvO6lGkrRsWrkevOVs/dNVJ8lKesfSWhXk0vkAAKTDtb85jTF+ST+WdLGkJkmrjDEPWms3HfKwf5B0r7X2v4wxCyU9Imm6WzPh+P1lU6sk6eIFr18FC/h9+uBZ03XDiqnitpUAAKTPzStkKyQ1WGt3WGsHJN0j6erDHmMlDd9bp1RSi4vzwAGPbdynU6aUqaY0702fyw34FPCzLREAgHS5+bdnraTGQ95vGvrYob4p6UZjTJMGr459xsV5cJxaQhFtaArrbSexRwwAACe5GWRHWryyh71/g6RfWWvrJF0u6S5jzJtmMsbcbIxZbYxZ3d7e7sKoSMXjQ8uVlwztHwMAAM5wM8iaJE055P06vXlJ8iOS7pUka+0LkvIkVR7+jay1t1trl1trl1dVVbk0Lkbyl037NKuqULOqirweBQCAjOJmkK2SNMcYM8MYkyvpekkPHvaYPZIukiRjzAINBhmXwMagUN+AXtzRydUxAABc4FqQWWvjkm6R9JikzRp8NeVGY8ytxpirhh72RUkfM8asl3S3pJustYcva2IMWLmlTYmkPXjcBQAAcI6rB0ZZax/R4Gb9Qz/2j4f8fpOks92cAc54bOM+1ZTkaXFtqdejAACQcTijACOKDCT0zNZ2XbywWj4OGgMAwHEEGUb0t23tisaS7B8DAMAlBBlG9JdNrSrJC+j0meVejwIAQEYiyHBM8URST25u1UULqpXDKfwAALiCu0DjiJJJq5d3deqel/eoqy+mty3kdH4AANxCkOENtrX26H/WNuuBtc1qCUdVkOvXDSum6qIFBBkAAG4hyHDQr57bqW8+tEl+n9G5cyr1lcvm6+KF1SrI5Y8JAABu4m9aHHTPqkYtrivVHTedpsqioNfjAACQNdilDUlScyiiLft69PbFk4gxAABOMIIMkqSVm1slib1iAAB4gCCDJOnJLW2aXlGgmZWFXo8CAEDWIcigvoG4nt/eoQvnV8sYbo0EAMCJRpBBz27br4F4Um9dMNHrUQAAyEoEGbRyS5uKgwEtn86tkQAA8AJBluWSSauVW9p07twq5Qb44wAAgBf4GzjLvdYSVltPvy5iuRIAAM8QZFnuyc1t8hnp/HkEGQAAXiHIstyTW1p16tQJKi/M9XoUAACyFkGWxfaFo3qtuVsXslwJAICnCLIs9lR9myTpovmczg8AgJcIsiz25OZW1U3I19zqIq9HAQAgqxFkWSoaS+jZhv26aP5ETucHAMBjAa8HgPvdD+N+AAAfV0lEQVRe2N6heDKpedXFqioOyhijF7Z3KBpL6kJuJg4AgOcIsgwX6hvQ+37+opJ28P0JBTmaV1Os7khcBbl+nTGT0/kBAPAaQZbh1jWGlLTSP1yxQH6fUf2+Hm3Z16NdHQd01ZLJCgb8Xo8IAEDWI8gy3PrGsIyR3nPaFBXn5Rz8uLWWvWMAAIwRbOrPcOsauzS7qugNMSaJGAMAYAwhyDKYtVbrm8I6ZUqZ16MAAIBjIMgyWGNnRJ0HBnTKVIIMAICxjCDLYGsbuyRJS+oIMgAAxjKCLIOtbwwrL8eneTXFXo8CAACOgSDLYOsau3Ty5FLl+PkxAwAwlvE3dYaKJZJ6raWbDf0AAIwDBFmG2rK3RwPxJBv6AQAYBwiyDLWuKSSJDf0AAIwHBFmGWrcnpMqiXNVNyPd6FAAAMAKCLEOta+zSkroyTuQHAGAcIMgyUHc0pu3tB9jQDwDAOEGQZaANjWFJ0hKCDACAcYEgy0Drhzf0E2QAAIwLBFkGWrsnpJlVhSrNz/F6FAAAkAKCLMNYa7WuMaRTOO4CAIBxgyDLMC3hqPb39nMgLAAA4whBlmHW7eFAWAAAxhuCLMOsbwop1+/TgkklXo8CAABSRJBlmHV7Qlo4uUS5AX60AACMF/ytnUHiiaRebQ5zICwAAONMwOsBMHrxRFK7Ovq0ZV+3tuzt0YbmsCKxhJayoR8AgHGFIBun/rJxnz57z1pFY0lJkt9nNKuqUO88tVbnz5vo8XQAACAdBNk49VR9m3L8Pv3LOxZp/qRizZ5YpGDA7/VYAABgFAiycap+X48WTirRtcvqvB4FAAAcJzb1j0PWWm1r7dXc6mKvRwEAAA4gyMahveGoevrjmltDkAEAkAkIsnFoa2uPJGnuxCKPJwEAAE4gyMahg0HGkiUAABmBIBuHtrb2qqo4qAmFuV6PAgAAHECQjUNbW3s0j6tjAABkDIJsnEkmB19hOaea/WMAAGQKgmycaeqKKBJLcIUMAIAMQpCNM8Mb+ucQZAAAZAyCbJypP/gKS5YsAQDIFATZOLOttUeTS/NUnJfj9SgAAMAhBNk4U9/aywn9AABkGIJsHIknktrezj0sAQDINATZOLK7s08D8SRBBgBAhiHIxpFtbOgHACAjEWTjSP2+Xhkjzeam4gAAZBSCbBzZ2tajKRMKVJAb8HoUAADgIIJsHNm6r4f9YwAAZCCCbJwYiCe1c/8B9o8BAJCBCLJxYuf+A4onreZxBhkAABmHIBsnDt7DciJBBgBApiHIxomtrT3y+4xmVhV6PQoAAHAYQTZObG3t0bSKAuXl+L0eBQAAOIwgGye2tvZqHq+wBAAgIxFk40A0ltDujgOaQ5ABAJCRCLJxoKGtV0krrpABAJChUgoyY8wfjDFXGGMIOA9sa+MelgAAZLJUA+u/JL1X0jZjzHeMMfNdnAmHqd/Xqxy/0fRKXmEJAEAmSinIrLVPWGvfJ+lUSbskPW6Med4Y8yFjTI6bA2Yza60eWNes+9c0as7EYuX4uUAJAEAmSvku1caYCkk3Snq/pLWSfivpHEkflHS+G8Nls817u/WNBzbq5V2dWlRbqu9cu8jrkQAAgEtSCjJjzP9Imi/pLklXWmv3Dn3q98aY1W4Nl43CfTF9//F63fXibpXm5+jb71ykdy+fIr/PeD0aAABwSapXyH5urX3k0A8YY4LW2n5r7XIX5spa1//3i6rf160bz5imL1w8V2UFuV6PBAAAXJbqpqR/PsLHXnByEEiRgYQ27+3WLRfO0a1Xn0yMAQCQJY55hcwYUyOpVlK+MWappOF1sxJJBS7PlnVawhFJ0vQKnloAALLJSEuWl0i6SVKdpO8f8vEeSX/v0kxZa28oKkmaXJbv8SQAAOBEOmaQWWvvlHSnMeZaa+0fTtBMWaslNHiFrJYgAwAgq4y0ZHmjtfY3kqYbY75w+Oettd8/wpdhlJpDERkjVZfkeT0KAAA4gUZashw+Gp579pwALaGIJhYHlRvgAFgAALLJSEuWPxv69Z9OzDjZrSUc0aRSlisBAMg2Iy1Z/vBYn7fWftbZcbJbSyiqhZNKvB4DAACcYCMtWa45IVNA1lq1hCJ664KJXo8CAABOsFReZYkToPPAgPrjSY68AAAgC420ZPkDa+3/McY8JMke/nlr7VWuTZZlWjiDDACArDXSkuVdQ7/+m9uDZLtmziADACBrjbRkuWbo12eMMbmS5mvwSlm9tXbgBMyXNYYPhZ1UyhlkAABkm5GukEmSjDFXSPqppO0avJ/lDGPMx621j7o5XDZpCUUUDPhUXsgNxQEAyDYpBZmkf5d0gbW2QZKMMbMk/UkSQeaQveGoasvyZYwZ+cEAACCjpHokfNtwjA3ZIanNhXmyVnMowoZ+AACy1Eivsnzn0G83GmMekXSvBveQvUvSKpdnyyotoYjOn1fl9RgAAMADIy1ZXnnI71slnTf0+3ZJE1yZKAv1xxNq6+nnChkAAFlqpFdZfuhEDZLNWsP9kjiDDACAbJXqqyzzJH1E0kmSDp7LYK39sEtzZZXhM8gmc2NxAACyUqqb+u+SVCPpEknPSKqT1OPWUNlmb3goyMo4gwwAgGyUapDNttZ+XdKBoftbXiFpkXtjZZfhQ2FZsgQAIDulGmSxoV9DxpiTJZVKmu7KRFmoORRVRWGu8nL8Xo8CAAA8kOrBsLcbYyZI+rqkByUVDf0eDmjhDDIAALJaSkFmrf350G+fkTTTvXGyU0sooplVhV6PAQAAPJLSkqUxpsIYc5sx5hVjzBpjzA+MMRVuD5cNrLVqCUU0iVdYAgCQtVLdQ3aPBm+VdK2k6yTtl/T7kb7IGHOpMabeGNNgjPnqET7/H8aYdUNvW40xoXSGzwTd0bgODCRUy5IlAABZK9U9ZOXW2m8d8v4/G2PecawvMMb4Jf1Y0sWSmiStMsY8aK3dNPwYa+3nD3n8ZyQtTXnyDMErLAEAQKpXyJ4yxlxvjPENvb1b0p9G+JoVkhqstTustQMavMp29TEef4Oku1OcJ2O8HmScQQYAQLYa6ebiPRq8mbiR9AVJvxn6lE9Sr6RvHOPLayU1HvJ+k6TTj/LPmSZphqSVKU2dQYaDjCVLAACy10j3siw+ju9tjvQtj/LY6yXdb61NHPEbGXOzpJslaerUqccx0tjTHIoqx29UWRT0ehQAAOCRVPeQyRhzlaRzh9592lr78Ahf0iRpyiHv10lqOcpjr5f06aN9I2vt7ZJul6Tly5cfLerGpZZQRDWlefL5jtSvAAAgG6R67MV3JH1O0qaht88NfexYVkmaY4yZYYzJ1WB0PXiE7z1P0gRJL6QzeKbYG45wU3EAALJcqlfILpd0irU2KUnGmDslrZX0pqMshllr48aYWyQ9Jskv6Q5r7UZjzK2SVltrh+PsBkn3WGsz6spXqlpCUZ0+o9zrMQAAgIdSXrKUVCapc+j3pal8gbX2EUmPHPaxfzzs/W+mMUNGiSeS2tcd5cgLAACyXKpB9m1Ja40xT2lws/65kr7m2lRZoq2nX4mkJcgAAMhyIwaZMcZIelbSGZJO02CQfcVau8/l2TIeZ5ABAAAphSCz1lpjzB+ttct0hE35GL2WcFQSp/QDAJDtUj2p/0VjzGmuTpKFhq+QTSrlChkAANks1T1kF0j6hDFml6QDGly2tNbaxW4Nlg1aQhGV5AVUnJfj9SgAAMBDqQbZZa5OkaVaQhGWKwEAwIj3ssyT9AlJsyW9KukX1tr4iRgsGzSHotzDEgAAjLiH7E5JyzUYY5dJ+nfXJ8pAA/GkXtrRoY7e/jd8nCtkAABAGnnJcqG1dpEkGWN+Iell90fKPA9vaNEX7l0vSZpaXqClU8u0qLZU4UhMkzjyAgCArDdSkMWGfzN0KySXx8lM+7oHj7f48iXz9GpTWC9s79AD6wbvsz69otDL0QAAwBgwUpAtMcZ0D/3eSMofen/4VZYlrk6XIcJ9MQUDPn36gtmSJGut9oaj2tF+QKfP5D6WAABku2MGmbXWf6IGyWShvphK818/2sIYo8ll+ewfAwAAklI/GBbHIRQZUFkBZ40BAIAjI8hOgHAkprL8XK/HAAAAYxRBdgKE+mIq5QoZAAA4CoLsBAhH3riHDAAA4FAE2QkwuGRJkAEAgCMjyFzWH0+obyDBpn4AAHBUBJnLwpHBs3VLC9jUDwAAjowgc1m4byjIWLIEAABHQZC5bPgKGXvIAADA0RBkLgsNXSFjDxkAADgagsxloYNXyNhDBgAAjowgc1mob0ASe8gAAMDREWQu647EZIxUnHfM+7gDAIAsRpC5LDR0Sr/PZ7weBQAAjFEEmctCfdw2CQAAHBtB5rIQt00CAAAjIMhcFo7EOKUfAAAcE0HmsnDfAFfIAADAMRFkLhve1A8AAHA0BJmLkkmr7kiMU/oBAMAxEWQu6umPK2k5FBYAABwbQeai8MH7WLKpHwAAHB1B5qJQhNsmAQCAkRFkLgoP31icPWQAAOAYCDIXhYaXLLlCBgAAjoEgc1Fo6ApZKVfIAADAMRBkLgr3sYcMAACMjCBzUTgSU36OX8GA3+tRAADAGEaQuSjUx6GwAABgZASZi7htEgAASAVB5qJwH0EGAABGRpC5KMx9LAEAQAoIMheFIgMqy+e2SQAA4NgIMhexqR8AAKSCIHNJNJZQfzypEvaQAQCAERBkLuE+lgAAIFUEmUtev48le8gAAMCxEWQuCQ3dNokrZAAAYCQEmUuGlyw5hwwAAIyEIHNJiCADAAApIshcEu5jUz8AAEgNQeaSUGRAfp9RUTDg9SgAAGCMI8hcEh66sbgxxutRAADAGEeQuSTUF1MZ+8cAAEAKCDKXhCMxlbJ/DAAApIAgc0moL8YrLAEAQEoIMpeEIyxZAgCA1BBkLgn1DaisgNsmAQCAkRFkLkgkrbqjcZYsAQBASggyF/REOaUfAACkjiBzQYhT+gEAQBoIMhcM38eSIAMAAKkgyFwQ6huQJJXms6kfAACMjCBzQTjCHjIAAJA6gswFYZYsAQBAGggyFwxv6ucKGQAASAVB5oJQX0xFwYBy/Dy9AABgZBSDC8IR7mMJAABSR5C5IBwZIMgAAEDKCDIXhPpibOgHAAApI8hcEIoQZAAAIHUEmQvYQwYAANJBkDnMWqtwX4xT+gEAQMoIModFYgkNJJIsWQIAgJQRZA47eEo/S5YAACBFBJnDOKUfAACkiyBz2MEgY8kSAACkiCBzWDgyIEkqY1M/AABIEUF2HH6/ao/++687FI0lDn7s4B4yrpABAIAUBbweYDy7bWWDmroi+uVzO/XFt83TNUtr2UMGAADSRpAdh2gsqdOmT1B/PKkv3rdeP392pyYWB5XjNyrI9Xs9HgAAGCdYsjwO/bGETq4t1R8/dbZ+eMNS9URjemZru0rzc2SM8Xo8AAAwTnCF7DhE4wnl5fjl8xldtWSyLjmpWr97aY9yA3QuAABIHUE2SomkVSxhlRd4fWkyGPDrQ2fP8HAqAAAwHnEpZ5SGX1mZl8NTCAAAjg81MUqvBxmb9wEAwPEhyEYpGk9K4goZAAA4ftTEKHGFDAAAOIUgG6XhIAsGCDIAAHB8CLJRisZYsgQAAM6gJkapnyVLAADgEIJslKJxggwAADiDIBslliwBAIBTqIlROvgqSzb1AwCA40SQjdLrV8gIMgAAcHwIslHi1kkAAMAp1MQosakfAAA4hSAbpeEly1w/TyEAADg+1MQo9ccSyg345PMZr0cBAADjHEE2StFYQnkBnj4AAHD8KIpRisaS7B8DAACOIMhGKRpPEGQAAMARBNkoRWMJjrwAAACOoChGiSVLAADgFIJslAY39RNkAADg+BFkoxSNJxVkyRIAADiAohil/hib+gEAgDNcDTJjzKXGmHpjTIMx5qtHecy7jTGbjDEbjTG/c3MeJ0UJMgAA4JCAW9/YGOOX9GNJF0tqkrTKGPOgtXbTIY+ZI+lrks621nYZYya6NY/TorEkB8MCAABHuFkUKyQ1WGt3WGsHJN0j6erDHvMxST+21nZJkrW2zcV5HMU5ZAAAwCluBlmtpMZD3m8a+tih5kqaa4x5zhjzojHm0iN9I2PMzcaY1caY1e3t7S6Nmx7OIQMAAE5xsyiOdNdte9j7AUlzJJ0v6QZJPzfGlL3pi6y93Vq73Fq7vKqqyvFB02Wt5RwyAADgGDeDrEnSlEPer5PUcoTHPGCtjVlrd0qq12CgjWn98aQkEWQAAMARbgbZKklzjDEzjDG5kq6X9OBhj/mjpAskyRhTqcElzB0uzuSI/thgkAXZ1A8AABzgWlFYa+OSbpH0mKTNku611m40xtxqjLlq6GGPSeowxmyS9JSkL1trO9yaySnReEISV8gAAIAzXDv2QpKstY9IeuSwj/3jIb+3kr4w9DZuDF8hI8gAAIATWHMbhdevkPH0AQCA40dRjEI0NhRk3FwcAAA4gCAbhShLlgAAwEEE2SgcvELGkiUAAHAARTEKrwcZV8gAAMDxI8hGIXrwYFiePgAAcPwoilEYvkIWZFM/AABwAEE2Cv0sWQIAAAcRZKPw+qssefoAAMDxoyhGgU39AADASQTZKETjCfl9Rjl+nj4AAHD8KIpRiMaSygvw1AEAAGdQFaMQjSVYrgQAAI4hyEYhGksSZAAAwDEE2ShE4wkFeYUlAABwCFUxCv2xhPI4FBYAADiEIBuFwSVLnjoAAOAMqmIU2NQPAACcRJCNQjROkAEAAOcQZKPAkiUAAHASVTEKUTb1AwAABxFkoxCNJRVkyRIAADiEIBuF/liCJUsAAOAYqmIU2NQPAACcRJClKZG0iiUse8gAAIBjCLI0RWMJSWLJEgAAOIaqSNPrQcYVMgAA4AyCLE3ReFISV8gAAIBzqIo0cYUMAAA4jSBL03CQBdnUDwAAHEKQpSkaY8kSAAA4i6pIUz9LlgAAwGEEWZqicYIMAAA4iyBLE0uWAADAaVRFmg6+ypJN/QAAwCEEWZpev0JGkAEAAGcQZGni1kkAAMBpVEWa2NQPAACcRpClaXjJMhjgqQMAAM6gKtLUH0soGPDJGOP1KAAAIEMQZGmKxhIsVwIAAEcRZGmKxpJs6AcAAI6iLNIUjXOFDAAAOIsgS1M0luBQWAAA4CiCLE0sWQIAAKdRFmmKxhIKcoUMAAA4iCBLUzSeVJArZAAAwEGURZr6OfYCAAA4jCBLE+eQAQAApxFkaYrGksrjtkkAAMBBlEWaOIcMAAA4jSBL0+CSJU8bAABwDmWRBmut+uNJrpABAABHEWRpGEgkZa0IMgAA4CiCLA3RWFKSFGRTPwAAcBBlkYb+WEISV8gAAICzCLI0DF8hI8gAAICTCLI0ROPDV8h42gAAgHMoizREh5csubk4AABwEEGWBpYsAQCAGwiyNBy8QsaSJQAAcBBlkYYor7IEAAAuIMjSEI0PL1nytAEAAOdQFmkYvkIWZFM/AABwEEGWBg6GBQAAbiDI0vD6qyx52gAAgHMoizSwqR8AALiBIEtDNJ6Q32eU4+dpAwAAzqEs0hCNJZUX4CkDAADOoi7SEI0lWK4EAACOI8jSEI0lCTIAAOA4giwN0XhCQV5hCQAAHEZdpKE/llAeh8ICAACHEWRpGFyy5CkDAADOoi7SwKZ+AADgBoIsDdE4QQYAAJxHkKWBJUsAAOAG6iINUTb1AwAAFxBkaYjGkgqyZAkAABxGkKWhP5ZgyRIAADiOukgDm/oBAIAbCLIUJZJWsYRlDxkAAHAcQZaiaCwhSSxZAgAAx1EXKXo9yLhCBgAAnEWQpSgaT0riChkAAHAedZEirpABAAC3EGQpGg6yIJv6AQCAwwiyFEVjLFkCAAB3UBcp6mfJEgAAuIQgS1E0TpABAAB3EGQpYskSAAC4hbpI0cFXWbKpHwAAOIwgS9HrV8gIMgAA4CyCLEXcOgkAALiFukgRm/oBAIBbCLIUDS9ZBgM8ZQAAwFnURYr6YwkFAz4ZY7weBQAAZBiCLEXRWILlSgAA4AqCLEXRWJIN/QAAwBUURoqica6QAQAAdxBkKYrGEhwKCwAAXEGQpYglSwAA4BYKI0XRWEJBliwBAIALCLIUReNJ9pABAABXEGQp6o8llMehsAAAwAUURor6uUIGAABcQpClaPBgWJ4uAADgPAojRZzUDwAA3EKQpWjw2AuCDAAAOM/VIDPGXGqMqTfGNBhjvnqEz99kjGk3xqwbevuom/OkYnt7rz5wx8v604a96o8nJEnWWkXjgzcXBwAAcFrArW9sjPFL+rGkiyU1SVpljHnQWrvpsIf+3lp7i1tzpKslFFFDa48+/btXNKEgR+88tU7XLK2VteIKGQAAcIVrQSZphaQGa+0OSTLG3CPpakmHB9mY8pY5VfrbVy7U37a1697Vjfr1C7v0i2d3ShJXyAAAgCvcDLJaSY2HvN8k6fQjPO5aY8y5krZK+ry1tvHwBxhjbpZ0syRNnTrVhVHfyO8zOn/eRJ0/b6I6evv1v2ub9eTmNq2YUe76PxsAAGQfNy/5mCN8zB72/kOSpltrF0t6QtKdR/pG1trbrbXLrbXLq6qqHB7z2CqKgvroW2bq7pvP0OK6shP6zwYAANnBzSBrkjTlkPfrJLUc+gBrbYe1tn/o3f+WtMzFeQAAAMYkN4NslaQ5xpgZxphcSddLevDQBxhjJh3y7lWSNrs4DwAAwJjk2h4ya23cGHOLpMck+SXdYa3daIy5VdJqa+2Dkj5rjLlKUlxSp6Sb3JoHAABgrDLWHr6ta2xbvny5Xb16tddjAAAAjMgYs8Zau3ykx3GOAwAAgMcIMgAAAI8RZAAAAB4jyAAAADxGkAEAAHiMIAMAAPAYQQYAAOAxggwAAMBjBBkAAIDHCDIAAACPEWQAAAAeI8gAAAA8RpABAAB4jCADAADwGEEGAADgMYIMAADAYwQZAACAxwgyAAAAjxFkAAAAHjPWWq9nSIsxpl3S7lF+eaWk/Q6OA+fxMxrb+PmMffyMxjZ+PmOf0z+jadbaqpEeNO6C7HgYY1Zba5d7PQeOjp/R2MbPZ+zjZzS28fMZ+7z6GbFkCQAA4DGCDAAAwGPZFmS3ez0ARsTPaGzj5zP28TMa2/j5jH2e/Iyyag8ZAADAWJRtV8gAAADGnKwIMmPMpcaYemNMgzHmq17PA8kYM8UY85QxZrMxZqMx5nNDHy83xjxujNk29OsEr2fNZsYYvzFmrTHm4aH3ZxhjXhr6+fzeGJPr9YzZzBhTZoy53xizZei/pTP5b2hsMcZ8fuh/414z/7+9+wnRqgrjOP590KQ0SkqK0sIEqSQoJUIqQqxFlmREUWEkYrQJrEii2lSLFkH2j0QItRTCEJNy1caC2iRlQkVuwkKnTIVSo0CTnhbnDL0MM7tmzsX7/cAw7zlzFw8cnnd+c88980ZsjYiz7aO2ImJTRByJiO8G5kbtmyjerPnhm4hYMF51nfGBLCImAeuAJcA84MGImNe2KgGngacy82pgIfBYXZdngF2ZORfYVcdq53Fg38D4ZeC1uj6/A6uaVKVhbwAfZ+ZVwLWUtbKHOiIiZgKrgesz8xpgEvAA9lFr7wK3j5gbq2+WAHPr16PA+vEq6owPZMANwA+ZuT8zTwHvA8sa19R7mXkoM7+ur/+g/CKZSVmbzfWyzcDdbSpURMwC7gQ21HEAi4Ht9RLXp6GIOA+4BdgIkJmnMvMY9lDXTAbOiYjJwFTgEPZRU5n5GfDbiOmx+mYZsCWLL4DpEXHJeNTVh0A2Ezg4MB6qc+qIiJgNzAd2Axdn5iEooQ24qF1lvfc68DTwTx1fCBzLzNN1bC+1NQc4CrxTt5U3RMQ07KHOyMyfgVeAA5QgdhzYg33URWP1zYRliD4EshhlzqOlHRER5wIfAE9k5onW9aiIiKXAkczcMzg9yqX2UjuTgQXA+sycD/yJ25OdUp9DWgZcAVwKTKNsgY1kH3XXhL3v9SGQDQGXDYxnAb80qkUDIuIsShh7LzN31OnDw7eD6/cjrerruZuAuyLiJ8o2/2LKHbPpdesF7KXWhoChzNxdx9spAc0e6o7bgB8z82hm/g3sAG7EPuqisfpmwjJEHwLZl8DceqplCuWByp2Na+q9+jzSRmBfZr468KOdwIr6egXw0UTXJsjMZzNzVmbOpvTMJ5m5HPgUuLde5vo0lJm/Agcj4so6dSvwPfZQlxwAFkbE1PqeN7xG9lH3jNU3O4GH62nLhcDx4a3N/1sv/jFsRNxB+et+ErApM19qXFLvRcTNwOfAt/z3jNJzlOfItgGXU97M7svMkQ9fagJFxCJgTWYujYg5lDtmFwB7gYcy82TL+vosIq6jHLqYAuwHVlL+0LaHOiIiXgTup5ws3ws8QnkGyT5qJCK2AouAGcBh4HngQ0bpmxqk36KcyvwLWJmZX41LXX0IZJIkSV3Why1LSZKkTjOQSZIkNWYgkyRJasxAJkmS1JiBTJIkqTEDmaTeioiMiLUD4zUR8ULDkiT1lIFMUp+dBO6JiBmtC5HUbwYySX12GngbeLJ1IZL6zUAmqe/WAcsj4vzWhUjqLwOZpF7LzBPAFmB161ok9ZeBTJLKZ92uAqa1LkRSPxnIJPVe/fDtbZRQJkkTzkAmScVawNOWkpqIzGxdgyRJUq95h0ySJKkxA5kkSVJjBjJJkqTGDGSSJEmNGcgkSZIaM5BJkiQ1ZiCTJElqzEAmSZLU2L8HbypBPpt6agAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, list_results)\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('Probability')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- As we can see from the plot, the probabilities seem to only change for **every other $N$ value**\n",
    "    - *Why?*\n",
    "       - This is because you need two flips to tie\n",
    "           - e.g. $P(\\text{Tied at some point in 3 flips}) = P(\\text{Tied at some point in 2 flips})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- First, let's assume for simplicity that **$N$ is even**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Then, there is some $n$ such that $N=2n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Now, let's assume that of the $N$ flips, A wins $x$ of them\n",
    "    - There are three cases we need to consider:\n",
    "        1. $x<n$ i.e. A wins less than half of the flips\n",
    "        2. $x=n$ i.e. A wins exactly half the flips\n",
    "        3. $x>n$ i.e. A wins more than half the flips"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Scenario 1\n",
    "\n",
    "- This is essentially the ballot box problem, so:\n",
    "\n",
    "$$\n",
    "P(\\text{Tie at some point | Scenario 1}) = 2\\cdot \\frac{x}{N}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Scenario 2\n",
    "\n",
    "- Since they get the exact number of flips, they **must** tie so:\n",
    "\n",
    "$$\n",
    "P(\\text{Tie at some point | Scenario 2}) = 1\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Scenario 3\n",
    "\n",
    "- This again the ballot box problem, but reversed, so:\n",
    "\n",
    "$$\n",
    "P(\\text{Tie at some point | Scenario 1}) = 2\\cdot \\frac{N-x}{N}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- For each possible $x$ value, there are $\\binom{N}{x}$ different paths with $x$ wins for A, and we know there are $2^{N}$ total possible paths for $N$ flips so\n",
    "\n",
    "$$\n",
    "P(x\\text{ wins for A}) = \\frac{\\binom{N}{x}}{2^{N}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- So, we can combine all these to get (note: let $T$ represent the event that the two tie at some point in $N$ flips):\n",
    "\n",
    "$$\n",
    "P(T) = P(T \\cap x=0) + P(T \\cap x=1) + ... + P(T \\cap x=N-1) + P(T \\cap x=N)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "=\\frac{2}{2^{N}} \\left ( \\binom{N}{0}\\frac{0}{N} +  \\binom{N}{1}\\frac{1}{N} + ... + \\binom{N}{N-1}\\frac{N-(N-1)}{N} + \\binom{N}{N}\\frac{N-N}{N} \\right )\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- We note that the numerator for the term where $x=i$ is the same as the one where $x=N-i$\n",
    "- We also note that $\\binom{N}{x} = \\binom{N}{N-x}$\n",
    "    - Combining these two, we can convert this expression to:\n",
    "    \n",
    "$$\n",
    "P(T) = \\frac{2}{2^{N}}\\left ( 2\\cdot\\sum_{x=0}^{x=n-1}\\binom{N}{x}\\frac{x}{N} + \\binom{N}{n}\\right )\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- I haven't written it out, but apparently this simplifies to:\n",
    "\n",
    "$$\n",
    "P(T) = \\frac{2}{2^{N}}\\left (2^{N-1} - \\binom{N-1}{n} \\right ) = 1 - \\frac{\\binom{N-1}{n}}{2^{N-1}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Therefore, the probability that they never tie in the $N$ flips is equal to:\n",
    "\n",
    "$$\n",
    "P(\\text{Never Tie}) = 1 - P(T) = \\frac{\\binom{N-1}{n}}{2^{N-1}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Note: apparently, this expression can be rewritten as\n",
    "\n",
    "$$\n",
    "P(\\text{Never Tie}) = \\binom{N}{n}\\frac{1}{2^{N}} = \\binom{N}{N/2}\\frac{1}{2^{N}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Since when $N$ is odd, the probability doesn't change, we can use\n",
    "\n",
    "$$\n",
    "P(\\text{Never Tie}) = \\binom{N-1}{(N-1)/2}\\frac{1}{2^{N-1}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_______\n",
    "\n",
    "- Let's compare this value to our curve we simulated"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.special import comb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "list_probabilities = []\n",
    "for N in x:\n",
    "    if N % 2 == 0:\n",
    "        P = 1-comb(N, N/2)/(2**N)\n",
    "    list_probabilities.append(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, list_results, label = 'Simulated')\n",
    "plt.plot(x, list_probabilities, label = 'Calculated')\n",
    "plt.legend()\n",
    "plt.xlabel('N')\n",
    "plt.ylabel('Probability')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Eyyyyy"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}