probability-bernoulli-with-python.ipynb

{
  "cells": [
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "nh__WW5eAGue"
      },
      "source": [
        "# Introduction to Probability\n",
        "\n",
        "*The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates the impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (\"heads\" and \"tails\") are both equally probable; the probability of \"heads\" equals the probability of \"tails\"; and since no other outcomes are possible, the probability of either \"heads\" or \"tails\" is 1/2 (which could also be written as 0.5 or 50%).* --- [Source](https://en.wikipedia.org/wiki/Probability)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "EuF3SZnUAGue"
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import pandas as pd"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "LlAJdHFgAGue"
      },
      "source": [
        "## Example: Tossing Coins\n",
        "\n",
        "Consider the example of tossing a fair coin as the ground for our vocabulary and symbols. We will say the probability of getting a head is the total number of ways for this event to happen divided by the number of all possible outcomes.\n",
        "\n",
        "**EXAMPLE I**: A Fair Coin\n",
        "\n",
        "With one fair coin, we have either heads or tails. The outcome of interest is either heads or tails. We would say the probability of each is $\\frac{1}{2}$:\n",
        "\n",
        "$$P(\\text{head}) = \\frac{1}{2} \\quad P(\\text{tails}) = \\frac{1}{2}$$\n",
        "\n",
        "Note that these probabilities sum to 1."
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "n-h85_v3AGue"
      },
      "source": [
        "**EXAMPLE II**: An Unfair Coin\n",
        "\n",
        "An example of an unfair coin would be anything that does not have equal probability.  For example:\n",
        "\n",
        "$$P(\\text{head}) = \\frac{6}{10} \\quad P(\\text{tails}) = \\frac{4}{10}$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 333
        },
        "id": "qpjlVffZAGue",
        "outputId": "a0ce31d8-365d-4cb9-c67d-9aa812661ae3"
      },
      "outputs": [
        {
          "data": {
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",
            "text/plain": [
              "<Figure size 1440x360 with 2 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light",
            "tags": []
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "import scipy.stats as stats\n",
        "fig, ax = plt.subplots(1, 2, figsize = (20, 5))\n",
        "fair_coin = stats.bernoulli(.5).pmf([0, 1])\n",
        "ax[0].bar(['heads', 'tails'], fair_coin, color=['gray', 'white'], edgecolor = 'black')\n",
        "ax[0].set_title('Probabilities for a Fair Coin');\n",
        "\n",
        "unfair_coin = stats.bernoulli(.4).pmf([0, 1])\n",
        "ax[1].bar(['heads', 'tails'], unfair_coin, color = ['gray', 'white'], edgecolor = 'black')\n",
        "ax[1].set_title('Probabilities for an Unfair Coin');"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "Bj3lEIqSAGuf"
      },
      "source": [
        "### Probability Mass Function\n",
        "\n",
        "We can describe this kind of event as a function. Specifically, a situation in which there are two possible outcomes with a probability attached to each. This is called a *Bernoulli* distribution, and the functional form is:\n",
        "\n",
        "$$f(k;p)=pk+(1-p)(1-k)\\quad {\\text{for }}k\\in \\{0,1\\}$$"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "tyEv6UmsAGuf"
      },
      "source": [
        "In our example of an unfair coin, if we consider heads as 0 and tails as 1 ($k$), we would have:\n",
        "\n",
        "$$f(0, .4) = .4*0 + (1 - .4)(1 - 0)$$"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "THV-KaywAGuf"
      },
      "source": [
        "Which equals 0.6. Similarly, the probability of a tails ($k = 0$):\n",
        "\n",
        "$$f(1, .4) = .4 * 1 + (1 - 1)(1 - 1)$$\n",
        "\n",
        "Which equals 0.4"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "19KM3ng6AGug"
      },
      "outputs": [],
      "source": [
        "def bernoulli(k, p):\n",
        "    return p*k + (1-p)*(1 - k)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "qTUJ-D5yAGug",
        "outputId": "131e3fc5-2539-4e25-f381-ffc61d098f03"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "0.6"
            ]
          },
          "execution_count": 24,
          "metadata": {
            "tags": []
          },
          "output_type": "execute_result"
        }
      ],
      "source": [
        "bernoulli(0, 0.4)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "MymmHDp6AGug",
        "outputId": "6bd809e7-bee1-4361-8f31-7a176e583de3"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "0.4"
            ]
          },
          "execution_count": 25,
          "metadata": {
            "tags": []
          },
          "output_type": "execute_result"
        }
      ],
      "source": [
        "bernoulli(1, 0.4)"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "gqwMxdVTAGug"
      },
      "source": [
        "### PROBLEMS\n",
        "\n",
        "1. Suppose a bag contains 8 Red Marbles and 3 Blue Marbles.\n",
        " - What is the probability of choosing a red ball from a random draw?\n",
        " - And the probability of a blue ball?\n",
        "2. Suppose a bag contains 4 Red Marbles, 7 Yellow Marbles, and 2 Blue Marbles.\n",
        " - What is the probability of selecting a red *or* yellow marble?\n",
        " - What is the probability of selecting a yellow *or* blue marble?\n",
        " - How many different combinations of two colors of marbles can be selected from the bag?\n",
        "3. Suppose a bag contains 10 Red and 7 Blue marbles.\n",
        " - Suppose you select one marble from the bag, what is the probability this is red?\n",
        " - Suppose you select a blue marble, place the marble back in the bag, and select another marble. What is the probability the second marble is blue?\n",
        " - Suppose you select a blue marble, do not replace the marble and select another marble. What is the probability the second marble is blue?"
      ]
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "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": "3.8.2"
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}