Probability.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probability Theory:\n",
    "\n",
    "Probability theory is a field in mathematics that study the caluclation of probability of occuring of events.  \n",
    "\n",
    "Example, the probability of tossing a coin and get Head is 50%.  \n",
    "\n",
    "## Simulate tossing a coin using Monte Carlo Simulator:  \n",
    "\n",
    "Let us use [Monte Carlo Simulator](https://www.investopedia.com/terms/m/montecarlosimulation.asp) to simulate tossing a coin 1000 times and calculate how much Head we get, and repeat that exercise 10 times.\n",
    "\n",
    "### 1. Numpy and random genertion:\n",
    "\n",
    "Real random numbers are difficult to produce, so in practice, we use pseudo-random numbers. Pseudo-random numbers are sufficiently random for most intents and purposes, except for some very exceptional instances, such as very accurate simulations. \n",
    "\n",
    "Numpy random number generation is based on the [Mersenne Twister algorithm](https://en.wikipedia.org/wiki/Mersenne_twister).\n",
    "\n",
    "### 2. Simulate tossing a coin (K) number of times:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "def generate_random_tosses(times, repeat):\n",
    "    random_array = np.random.rand(times, repeat) # this will generate random of float between 0 and 1\n",
    "    # to simulate tossing coin we convert any greater than 0.5 to +1 and anything else to -1\n",
    "    coin_tosses = (random_array > 0.5) * 2 - 1\n",
    "    head_counts = np.sum(coin_tosses, axis = 0) # find the sum where Head = (+1) and Tail = (-1)\n",
    "    return head_counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "results:  [ -2 -26  12 -18   0  16  86  -2   2 -44]\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "times = 1000\n",
    "repeat = 10\n",
    "\n",
    "tosses_results = generate_random_tosses(times, repeat)\n",
    "print('results: ', tosses_results)\n",
    "\n",
    "plt.figure(figsize=[10,2])\n",
    "plt.hist(tosses_results);\n",
    "plt.xlim([-times,times])\n",
    "plt.xlabel(\"sum\")\n",
    "plt.ylabel(\"count\")\n",
    "plt.title(\"Histogram of coin flip sum when flipping a fair coin %d times\"%times)\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Monte Carlo Simulator:\n",
    "Using the above code, we are going to repeat the exercise many times while augmenting the value: \"**`times`**\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 936x252 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from math import sqrt\n",
    "plt.figure(figsize=[13,3.5])\n",
    "for j in range(2,5):\n",
    "    k=10**j\n",
    "    counts=generate_random_tosses(times=k,repeat=100)\n",
    "    plt.subplot(130+j-1)\n",
    "    plt.hist(counts,bins=10);\n",
    "    d=4*sqrt(k)\n",
    "    plt.plot([-d,-d],[0,30],'r')\n",
    "    plt.plot([+d,+d],[0,30],'r')\n",
    "    plt.grid()\n",
    "    plt.title('%d flips, bound=+-%6.1f'%(k,d))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us repeat the previous one, but now let us draw on Axis X of the plot, the whole scale of K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 936x252 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=[13,3.5])\n",
    "for j in range(2,5):\n",
    "    k=10**j\n",
    "    counts=generate_random_tosses(k,100)\n",
    "    plt.subplot(130+j-1)\n",
    "    plt.hist(counts,bins=10);\n",
    "    plt.xlim([-k,k])\n",
    "    d=4*sqrt(k)\n",
    "    plt.plot([-d,-d],[0,30],'r')\n",
    "    plt.plot([+d,+d],[0,30],'r')\n",
    "    plt.grid()\n",
    "    plt.title('%d flips, bound=+-%6.1f'%(k,d))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}