Lab1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Elementary Notions In Probability\n",
    "\n",
    "A quick reminder on probability theory: https://youtu.be/a-IVT3AxwSg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sample space $\\Omega$\n",
    "\n",
    "> A _sample space_ is a set with elements describing the outcomes (also called elementary events) of the experiment in which we are interested.\n",
    "\n",
    "Examples:\n",
    "* Tossing a coin:\n",
    "  $\\Omega = \\{Head, Tail\\}$\n",
    "* Number of bees in Europe: $\\Omega = \\mathbb{N}$\n",
    "* Guessing the month when someone is born:\n",
    "  $\\Omega = \\{Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec\\}$\n",
    "  \n",
    "In Python, these can be represented simply as [sets](https://docs.python.org/3/library/stdtypes.html?highlight=set#set)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Omega: {'H', 'T'}\n"
     ]
    }
   ],
   "source": [
    "omega = {'H', 'T'} # Or use the set() constructor\n",
    "print('Omega:',omega)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Print the sample space corresponding to a single die throw, using [`range()`](https://docs.python.org/3/library/functions.html#func-range)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expected result: `{1, 2, 3, 4, 5, 6}`\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Event $A \\subseteq \\Omega$\n",
    "\n",
    "> An _event_ is a subset of the sample space.\n",
    "\n",
    "Examples:\n",
    "* The certain event: $\\Omega$\n",
    "* The impossible event: $\\emptyset$\n",
    "* Result of a coin toss: $A = \\{Head\\}$\n",
    "* Knowing the someone is born in a summer month: $A=\\{Jun, Jul, Aug\\}$\n",
    "\n",
    "### Operations\n",
    "\n",
    "Generating a random event from a given sample space is effectively random sampling without replacement, so the `sample` function from the [random](https://docs.python.org/3/library/random.html) library can be used:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Omega: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}\n",
      "A: {0, 5, 7}\n",
      "B: {0, 1, 2, 3, 9}\n"
     ]
    }
   ],
   "source": [
    "from random import sample\n",
    "omega = set(range(10))\n",
    "A = set(sample(omega, k=3))\n",
    "B = set(sample(omega, k=5))\n",
    "print(\"Omega:\", omega)\n",
    "print(\"A:\", A)\n",
    "print(\"B:\",B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since events are sets, we can operate with events using the regular set operations: `union`, `intersection`, `difference`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{1, 2, 3}\n",
      "{2}\n",
      "{1}\n"
     ]
    }
   ],
   "source": [
    "A = {1, 2}\n",
    "B = {2, 3}\n",
    "omega = {1,2,3,4}\n",
    "print(A.union(B))        # Or A|B\n",
    "print(A.intersection(B)) # Or A&B\n",
    "print(A.difference(B))   # Or A-B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can visualise these operations using the [matplotlib_venn](https://pypi.org/project/matplotlib-venn/) library:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 400x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from tools.venn import A, B, omega, plot_venn\n",
    "plot_venn(omega)   # Print the entire space\n",
    "plot_venn(omega-A) # Print not A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Using the `plot_venn()` function and the existing `A`, `B` and `omega` variables, show that $A - B = A \\cap \\neg B$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from tools.venn import A, B, omega, plot_venn\n",
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Disjoint Events\n",
    "Events A and B are called _disjoint_ or _mutually exclusive_ if A and B have no outcomes in common. In set notation: $A \\cap B = \\emptyset$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Determine if the events $A$ and $B$ are disjoint:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from random import sample, seed\n",
    "seed(5)\n",
    "A = set(sample(range(1000), k=50))\n",
    "B = set(sample(range(1000), k=50))\n",
    "# Code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expected result: no, they have elements {186, 707} in common.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability\n",
    "\n",
    "> A _probability function_ $P$ on a finite sample space $\\Omega$ assigns to  each event $A$ in $\\Omega$ a number $P(A)$ in $[0,1]$ such that:\n",
    "> 1. $P(\\Omega)=1$ and\n",
    "> 1. $P(A \\cup B) = P(A)+P(B)$ if $A$ and $B$ are disjoint.\n",
    "\n",
    "The number $P(A)$ is called the probability that $A$ occurs.\n",
    "\n",
    "Consequently, if we have a finite space **and all outcomes are equally likely**:\n",
    "\n",
    "$P(A) = \\frac{\\mbox{\\#favourable outcomes}}{\\mbox{\\#all outcomes}} = \\frac{|A|}{|\\Omega|}$\n",
    "\n",
    "Useful properties resulting from the definition:\n",
    "1. $P(\\emptyset) = 0$\n",
    "1. $P(\\neg A) = 1-P(A)$\n",
    "1. $A \\subseteq B \\Rightarrow P(A) \\leq P(B)$\n",
    "1. $P(A \\setminus B) = P(A) - P(A \\cap B)$\n",
    "1. $P(A \\cup B) = P(A) + P(B) - P(A \\cap B)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Prove the first property: $P(\\emptyset) = 0$ using only the definition.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 1\n",
    "\n",
    "What is the probability that a die will turn an even number?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Omega: {1, 2, 3, 4, 5, 6}\n",
      "A: {2, 4, 6}\n",
      "P(A): 0.5\n"
     ]
    }
   ],
   "source": [
    "from tools.stats import probability\n",
    "\n",
    "omega = set(range(1,7))\n",
    "print('Omega:', omega)\n",
    "A = set(a for a in omega if a % 2 == 0)\n",
    "print('A:', A)\n",
    "print('P(A):', probability(A, omega))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2\n",
    "\n",
    "If not all outcomes are equally likely, we can use a weighted sum instead of simply counting the outcomes. The `probability_weighted` function does just that. \n",
    "\n",
    "If we have an unfair coin with a 75% chance of showing 'Head':"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.75"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from tools.stats import probability_weighted, WeightedOutcome\n",
    "from dataclasses import dataclass\n",
    "\n",
    "@dataclass(frozen=True)\n",
    "class Outcome(WeightedOutcome):\n",
    "    side: str\n",
    "        \n",
    "omega = set([Outcome(side='Head', weight=0.75), \n",
    "            Outcome(side='Tail', weight=0.25)])\n",
    "A = set(o for o in omega if o.side == 'Head')\n",
    "probability_weighted(A, omega)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Products of sample spaces\n",
    "\n",
    "Often in practice, more than one experiment is considered at a time. For instance, if we throw a coin two times, we would have two sample spaces $\\Omega_2 = \\Omega_1 = \\{H, T\\}$.\n",
    "\n",
    "We can treat both trows as a single event with the sample space \n",
    "\n",
    "$\\Omega = \\Omega_1 \\times \\Omega_2 = \\{(H,H), (H,T), (T,H), (T,T)\\}$\n",
    "\n",
    "or generally\n",
    "\n",
    "$\\Omega = \\Omega_1 \\times \\Omega_2 = \\{(\\omega_1, \\omega_2): \\omega_1 \\in \\Omega_1, \\omega_2 \\in \\Omega_2\\}$\n",
    "\n",
    "This corresonds to a cartesian product, so in Python it can be achieved throught the [product](https://docs.python.org/3/library/itertools.html#itertools.product) function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{('b', 2), ('a', 1), ('b', 1), ('a', 2)}\n"
     ]
    }
   ],
   "source": [
    "from itertools import product\n",
    "A = set(product(['a','b'], [1, 2]))\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "What is the probability that by throwing a coin two times we will get tails at least once?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Omega: {('T', 'T'), ('T', 'H'), ('H', 'T'), ('H', 'H')}\n",
      "A: {('T', 'H'), ('H', 'T'), ('T', 'T')}\n",
      "P(A)= 0.75\n"
     ]
    }
   ],
   "source": [
    "from itertools import product\n",
    "from tools.stats import probability\n",
    "omega_1=set(['H', 'T'])\n",
    "omega_2=set(['H', 'T'])\n",
    "omega=set(product(omega_1, omega_2))\n",
    "print('Omega:', omega)\n",
    "A = set(a for a in omega if 'T' in a)\n",
    "print('A:', A)\n",
    "print('P(A)=', probability(A, omega))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "What is the probability that a family with two children will have two girls?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from itertools import product\n",
    "from tools.stats import probability\n",
    "omega_1={'B', 'G'} # First child\n",
    "omega_2={'B', 'G'} # Second child\n",
    "# Your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expected result: 0.25\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conditional probabilities\n",
    "\n",
    "> The _conditional probability of A given B_ is defined as:\n",
    "> \n",
    "> $P(A|B)= \\frac{P(A\\cap B)}{P(B)}$\n",
    "\n",
    "Also called _posterior probability of A, given B_.\n",
    "\n",
    "Useful properties resulting from the definition:\n",
    "1. The multiplication rule:\n",
    "\n",
    "  $P(A \\cap B) = P(A|B)P(B) = P(B|A)P(A)$ \n",
    "  \n",
    "1. The chain rule:\n",
    "\n",
    "  $P(A_1 \\cap A_2 ... \\cap A_n) = P(A_1)P(A_2|A_1)...P(A_n|A_1,A_2,...,A_{n-1})$\n",
    "  \n",
    "1. The law of total probability (for two events):\n",
    "\n",
    "  $P(A) = P(A|B)P(B)+P(A| \\neg B)P(\\neg B)$\n",
    "  \n",
    "1. The law of total probability (for multiple events):\n",
    "\n",
    "  $P(A) = P(A|B_1)P(B_1)+P(A|B_2)P(B_2)+...+P(A|B_n)P(B_n)$ where $B_1,...B_n$ are disjoint events such that $A \\subseteq  B_1 \\cup ... \\cup B_n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "What is the probability that the current month has 31 days knowning that it's summer? We can model the problem using events like this:\n",
    "* $\\Omega$ is the sample spece of all months: $\\Omega = \\{Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec\\}$\n",
    "* $A$ is the event \"month with 31 days\": $A=\\{Jan, Mar, May, Jul, Aug, Oct, Dec\\}$\n",
    "* $B$ is the event \"month is a summer month\": $B=\\{Jun, Jul, Aug\\}$\n",
    "* We are interested in finding $P(A|B)$.\n",
    "\n",
    "$P(A|B)= \\frac{P(A\\cap B)}{P(B)} = \\frac{|\\{Jul, Aug\\}|}{|\\{Jun, Jul, Aug\\}|}=\\frac{2}{3}$\n",
    "\n",
    "Or, using Python and the `probability` function defined above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6666666666666666\n",
      "0.6666666666666666\n"
     ]
    }
   ],
   "source": [
    "from tools.stats import probability\n",
    "omega = set(['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun', \n",
    "             'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'])\n",
    "A = set(['Jan', 'Mar', 'May', 'Jul', 'Aug', 'Oct', 'Dec'])\n",
    "B = set(['Jun', 'Jul', 'Aug'])\n",
    "res1 = probability(A&B, omega) / probability(B, omega)\n",
    "res2 = probability(A&B, B) # We can just repace omega with B\n",
    "print(res1)\n",
    "print(res2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**Give it a try!**\n",
    "\n",
    "A bowl contains $w$ white balls and $b$ black balls. \n",
    "\n",
    " 1. We first extract one of these, at random.\n",
    " 2. We then place it back, together with $d$ other balls of the same colour.\n",
    " 3. We take out another ball at random.\n",
    "\n",
    "Prove that the probablity that the second ball is white, does not depend on $d$.\n",
    "\n",
    "---\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayes' Theorem\n",
    "\n",
    "![Bayes Theorem neon sign at HP Authonomy](img/Bayes_Theorem.jpg)\n",
    "\n",
    "Bayes' theorem can be interpreted as linking the degree of belief in a proposition before and after accounting for evidence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Give it a try!**\n",
    "\n",
    "Prove Bayes' theorem using the definition of conditional probabilities.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Independent events\n",
    "\n",
    "> An event A is called _independent of B_ if $P(A|B)=P(A)$\n",
    "\n",
    "Useful properties:\n",
    "1. Two events $A$ and $B$ are independent if $P(A \\cap B) = P(A)P(B)$\n",
    "1. Conditionally independent events: $P(A\\cap B\\ |\\ C)=P(A|C)P(B|C)$"
   ]
  }
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