Paradigms of Probability.ipynb

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   "source": [
    "Let's discuss the three Paradigms of Probability\n",
    "================================================\n",
    "\n",
    "###### we are going to use a 6 sided Dice as our assistant to understand them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
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   "outputs": [],
   "source": [
    "from random import choice\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "class Dice():\n",
    "    \"\"\"\"Initiate a N sided dice\"\"\"\n",
    "    def __init__(self, n):\n",
    "        # python excludes the last number thus n+1\n",
    "        self.sides = range(1, n+1)\n",
    "    \n",
    "    def roll(self):\n",
    "        return choice(self.sides)"
   ]
  },
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   "source": [
    "## 1. Classical"
   ]
  },
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   "source": [
    "Classical probability is definded by \n",
    "\n",
    "        favourable outcomes by total number of outcomes.\n",
    "        \n",
    "Fairly easy to compute.\n",
    "\n",
    "***Probability of getting a 4 = 1/6***"
   ]
  },
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   "source": [
    "## 2. Frequentist\n",
    "\n",
    "Frequentists measure the probability of an event happening based on the number of times that event has occured from the total obeserved frequency\n",
    "\n",
    "So, lets roll the 6 sided dice 10 times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "outputs": [],
   "source": [
    "# Emulate Dice rolls\n",
    "six_sided_dice = Dice(6)\n",
    "num_rolls = 10\n",
    "\n",
    "rolls = [six_sided_dice.roll() for _ in range(num_rolls)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "Let's view the probability of all the sides of the dice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
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   "outputs": [],
   "source": [
    "def frequentist_prob(n, rolls):\n",
    "    \"\"\"Calculate the frequentist probability of an event of rolling n given m number of rolls.\"\"\"\n",
    "    return rolls.count(n)/len(rolls)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "freq_prob = {side:frequentist_prob(side, rolls) for side in six_sided_dice.sides} "
   ]
  },
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   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
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   "outputs": [
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pMwYPSZLUGYOHJEnqjMFDkiR1xuAhSZI6Y/CQJEmdMXhIkqTOGDwkSVJnDB6S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\nSVJnDB6SJKkzBg9JktQZg4ckSerM/w/MBnYzNsr0WAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18fbc422f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Lets plot it for simplicity\n",
    "def plot_prob_distribution(freq_prob):\n",
    "    \"\"\"Plot the probability distribution.\"\"\"\n",
    "    plt.bar(range(len(freq_prob)), freq_prob.values(), align='center')\n",
    "    plt.xticks(range(len(freq_prob)), list(freq_prob.keys()))\n",
    "    plt.xlabel(\"Sides\")\n",
    "    plt.ylabel(\"Probabilities\")\n",
    "    plt.show()\n",
    "\n",
    "plot_prob_distribution(freq_prob)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "#### Exercise 1. \n",
    "\n",
    "Try out with different number of rolls."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Solution here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Solution here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Solution Here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Bayesian\n",
    "\n",
    "Bayesian approach is also known as \"*subjective probability*\", its represented by the formula : -\n",
    "\n",
    "\n",
    "                P(A|B) = P(B|A)*P(A) / P(B)\n",
    "\n",
    "        - P(A|B) is interpreted as proabbility of event A given that event B occurs, similarly P(B|A).\n",
    "        - P(B|A)*P(A) can be interpreted as a 'update' that you do to your existing probability P(A) with the \n",
    "          likelihood P(B|A)\n",
    "          (P.S. don't worry if this doesn't make much sense rite now.)\n",
    "        \n",
    "The simplest way to interpret what this formula is doing is that its **Calculating the reverse of Conditional Probability.**\n",
    "\n",
    "But, we are interested in the **Diachronic Interpretation**, which means :- \n",
    "\n",
    "Given a hypothesis H, what is the probability \n",
    "\n",
    "    of the hypothesis *before* seeing some data P(H) and\n",
    "    \n",
    "What is the porbability \n",
    "    \n",
    "    of the hypothesis *after* seeing the data P(H|D)\n",
    "    \n",
    "    \n",
    "    H: hypothesis\n",
    "    D: Data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Exercise 2\n",
    "\n",
    "Think of Hypothesis as a beleif that you would have. \n",
    "      \n",
    "      \"I beleive I would reach the station in 30 minutes.\"\n",
    " \n",
    "and think of data as an event that occured that made you alter your beleif.\n",
    "\n",
    "      \"Google maps says 15 minutes to the station.\"\n",
    " \n",
    " **Now fill the block below**"
   ]
  },
  {
   "cell_type": "markdown",
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    "Hypothesis: *Write your beleif Here*\n",
    "\n",
    "Data: *Event here*"
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    "## Discuss"
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    "Lets come back to the dice, what should be our Problem Statement ?\n",
    "\n",
    "\"Given output of 10 dice rolls, what is the probability of getting a 4\""
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