33-Solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Relating the Birthday Pairings and Birthmate Problems\n",
    "\n",
    "- If $r$ persons compare birthdays in the pairing problem, the probability is $P_{r}$ that at least 2 have the same birthday\n",
    "- *What should $n$ be in the personal birthmate problem to make your probability of success approximately $P_{r}$?*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_____\n",
    "\n",
    "- If $r$-people are comparing birthdays, each person has $r-1$ people to compare birthdays to\n",
    "    - Therefore, there are $\\frac{r(r-1)}{2}$ comparisons that can be made (divided by 2 since we can't double count)\n",
    "\n",
    "- We want the single person to have approximately the same number of comparisons, so:\n",
    "\n",
    "$$\n",
    "n \\approx \\frac{r(r-1)}{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- **Recall**: we found that 23 people have approximately a 0.5 chance of having at least one common birthday, and it requires a single person about 253 people to find someone with the same birthday"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{23\\cdot22}{2} = 253 = n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- As we can see, out approximation is pretty good"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_____\n",
    "\n",
    "- Let's derive the $n$ for each $r$ and plot them"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "plt.rcParams['figure.figsize'] = 10, 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def p1(r):\n",
    "    prob = 1\n",
    "    for i in range(r):\n",
    "        prob *= ((365-i)/365)\n",
    "    return 1-prob"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def p2(n):\n",
    "    return 1-(364/365)**n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = range(1, 1000)\n",
    "list_n = []\n",
    "\n",
    "for r_i in r:\n",
    "    P_r = p1(r_i)\n",
    "    n = r_i\n",
    "    P_n = p2(n)\n",
    "    while P_n < P_r:\n",
    "        n += 1\n",
    "        P_n = p2(n)\n",
    "    list_n.append(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.994122660865348"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1(60)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999987833"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p2(10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2d64d0a8d30>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(r, list_n, label='Real')\n",
    "plt.plot(np.arange(1, 200), np.arange(1, 200)*(np.arange(1, 200)-1)/2, label='Approximated')\n",
    "plt.xlabel('r')\n",
    "plt.ylabel('n')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- The reason the real curve takes this shape is because after a while, the probability converges to 1 so it's the same value every time\n",
    "    - As we can see, the approximation is pretty good for small r"
   ]
  }
 ],
 "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": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}