add probability

Author: gkarwchan

Probability.ipynb

@@ -0,0 +1,110 @@
+{
+ "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 Python:  \n",
+    "\n",
+    "Let us simulate tossing a coin 1000 times and calculate how much Head we get, and repeat that exercise 10 times.\n",
+    "\n",
+    "### 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)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "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": 28,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "results:  [  8  22 -26  36 -54 -30  -6 -30  14 -20]\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 720x288 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=[10,4])\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": "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": "3.7.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

presentation.ipynb

@@ -1322,7 +1322,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.6.5"
+   "version": "3.7.0"
   }
  },
  "nbformat": 4,