ponder_and_prove_conditional_probability_and_Bayes_theorem.ipynb

{
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  "metadata": {
    "colab": {
      "name": "ponder_and_prove_conditional_probability_and_Bayes_theorem.ipynb",
      "private_outputs": true,
      "provenance": [],
      "collapsed_sections": [
        "3VZLl2b8byS7",
        "tSUqeusUb2nE",
        "muxeYgQQKGEU"
      ],
      "include_colab_link": true
    },
    "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": "3.9.0"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/ultimatejacob27/cse380-notebooks/blob/master/ponder_and_prove_conditional_probability_and_Bayes_theorem.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uas6fBg_cTMW"
      },
      "source": [
        "# **Ponder and Prove Conditional Probability and Bayes' Theorem**\n",
        "Due: Saturday, 13 February 2021, 11:59pm"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ndqbmxkZOjAq"
      },
      "source": [
        "# TODO Watch a Video on Bayes' Theorem"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gF3YCXyTP4aJ"
      },
      "source": [
        "[and making probability intuitive](https://www.youtube.com/watch?v=HZGCoVF3YvM), 15.75 minutes long."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uwXpPms8Mgb3"
      },
      "source": [
        "# TODO Apply Bayes’ Theorem"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lB2cIND7Mcu7"
      },
      "source": [
        "\n",
        "Automating the process of finding bugs in source code requires the use of heuristics, or pattern-matching on the code. There is no deep understanding of the code involved, hence such a tool cannot be perfect at detecting bugs. A certain bug-detection tool is correct 80% of the time and incorrect the rest of the time. This means that on average out of every 100 bugs, the tool will detect 80 of them, and fail to detect 20 of them. Also, out of 100 non-bugs, the tool will report 20 bugs and 80 non-bugs, on average. The tool is run on source code that is 10% buggy. That is, one line out of every ten contains a bug. Given that the tool detects a bug in some line of code, what is the probability that this line of code actually contains a bug?\n",
        "\n",
        "On average, for every 100 lines of code, there will be 10 bugs. 8 (0.8 * 10) real bugs will be reported, and 18 (0.2 * 90) fake bugs will be reported. So the probability of a reported bug being real is 0.444 (8/18)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sBD8DPkM0H1i"
      },
      "source": [
        "## TA Comments"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qlYNdbHa0KJB"
      },
      "source": [
        "Hello Jacob! The correct answer for this question is approximately 0.30769, or 30.769%. I'll include an explanation for this below."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Ys6kDEjO0b1V"
      },
      "source": [
        "# p(e) = .10\r\n",
        "# e = bug exists\r\n",
        "# f = bug found\r\n",
        "# p(f | e) = .80\r\n",
        "# p(!f | e) = .20\r\n",
        "# p(f | !e) = .20\r\n",
        "# p(!f | !e) = .80\r\n",
        "\r\n",
        "# p(e | f) = p(f | e) * p(e) / p(f)\r\n",
        "\r\n",
        "# p(f) = p(f | e) * p(e) + p(f | !e) * p(!e)\r\n",
        "\r\n",
        "# p(e | f) = p(f | e) * p(e) / (p(f | e) * p(e) + p(f | !e) * p(!e))\r\n",
        "\r\n",
        "# p(e | f) = (.8 * .1) / ((.8 * .1) + (.2 * .9))\r\n",
        "\r\n",
        "# p(e | f) = .08 / (.08 + .18)\r\n",
        "\r\n",
        "# p(e | f) = .08 / .26 = 0.30769230769230776\r\n",
        "\r\n",
        "# If the tool detects a bug in a line of code in the source code given,\r\n",
        "# the probability that the line of code actually contains a bug is 30.769%."
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fcvmte7xawPl"
      },
      "source": [
        "# TODO Compute Four Leaves"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "motP9549ZMxc"
      },
      "source": [
        "\n",
        "Using the information in the previous task, calculate to two-decimal-digit precision, the values that go at the leaves of the binary tree shown below:\n",
        "\n",
        "![fig-binary-tree-with-four-leaves.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JzJxbCRCZ01-"
      },
      "source": [
        "The value that goes at leaf number 1 is calculated by $P(H) \\cdot P(T|H)$. The other three values are calculated similarly.\n",
        "\n",
        "Hint: The four leaves correspond to the four quadrants of trutH versus tesT described in this brief introduction to [Bayes' Theorem](https://byui-cse.github.io/cse380-course/Bayes-Theorem.pdf). (Compare this to the video you watched.)\n",
        "\n",
        "### Put your answers here:\n",
        "1. $P(H) \\cdot P(T|H)$ = (.1)(.8) = 0.08\n",
        "2. $P(H) \\cdot P(!T|H)$ = (.1)(.2) = 0.02\n",
        "3. $P(H) \\cdot P(T|!H)$ = (.9)(.2) = 0.18\n",
        "4. $P(H) \\cdot P(!T|!H)$ = (.9)(.8) = 0.72\n",
        "\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "i2_e7Jgi1Fon"
      },
      "source": [
        "## TA Comments"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tCP7S2201HXR"
      },
      "source": [
        "You had the correct values but the incorrect labels. I fixed them for you, and I will not deduct any points."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EO4g44kGbGh6"
      },
      "source": [
        "# TODO Apply Your Probability Knowledge\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FaNSq3v0bQDI"
      },
      "source": [
        "There are three sheets of paper in a folder. One sheet has writing on both sides, one has writing on one side and is blank on the other side, and the third sheet of paper is blank on both sides. Without looking you randomly pull one sheet of paper out of the folder and see that it is blank on the side you're looking at. What is the probability that the other side is blank as well?\r\n",
        "\r\n",
        "0.5"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IO-kcGvi1Q6b"
      },
      "source": [
        "## TA Comments"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "VIaOHK9p1TAU"
      },
      "source": [
        "The correct answer for this question is 2/3, or 66.667%. I'll supply an explanation for why this is the case below."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mmc7IFM11aNH"
      },
      "source": [
        "# Papers:\r\n",
        "# : |------|------|  |------|------|  |------|------|\r\n",
        "# : |      |      |  |      |      |  |      |      |\r\n",
        "# : |  W0  |  W1  |  |  W2  |  B0  |  |  B1  |  B2  |\r\n",
        "# : |      |      |  |      |      |  |      |      |\r\n",
        "# : |------|------|  |------|------|  |------|------|\r\n",
        "\r\n",
        "# | looking at  | other side  | valid? | is blank? |\r\n",
        "# |-------------+-------------+--------+-----------|\r\n",
        "# | W_0         | W_1         | no     |xxxxxxxxxxx|\r\n",
        "# | W_1         | W_0         | no     |xxxxxxxxxxx|\r\n",
        "# | W_2         | B_0         | no     |xxxxxxxxxxx|\r\n",
        "# | B_0         | W_2         | yes    | no        |\r\n",
        "# | B_1         | B_2         | yes    | yes       |\r\n",
        "# | B_2         | B_1         | yes    | yes       |\r\n",
        "\r\n",
        "# There are 3 possible cases where you could start out by looking at a blank page.\r\n",
        "# Of those three, only 2 have a blank page on the other side. Thus, the probability\r\n",
        "# that the other side of a blank page is blank as well is 2/3, .66667, or 66.667%."
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3VZLl2b8byS7"
      },
      "source": [
        "# TODO Apply Bayes Theorem Generalized"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uwkiPKzibWto"
      },
      "source": [
        "Three different vegetables, turnips, eggplant and asparagus, make Bill sick sometimes. There is a 50% chance he will be sick if he eats turnips. If he eats eggplant, there is a 20% chance he will be sick. The probability that he will be sick when he eats asparagus is only 5%.\n",
        "\n",
        "Bill gets sick after dinner where he ate one of those three vegetables. What is the probability that he ate turnips?\n",
        "\n",
        "(Assume that for each of the three vegetables, there is a 1/3 chance he ate it.)\n",
        "\n",
        "50/65 = 0.769"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1WMLDSgp2VaB"
      },
      "source": [
        "## TA Comments"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ARnIfKGa2bLO"
      },
      "source": [
        "The correct answer is 0.66667, or 66.667% (2/3). I will provide an explanation below."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZcJ4lO_d2wNS"
      },
      "source": [
        "# p(t) = .333\r\n",
        "# p(e) = .333\r\n",
        "# p(a) = .333\r\n",
        "\r\n",
        "# p(s | t) = .5\r\n",
        "# p(s | e) = .2\r\n",
        "# p(s | a) = .05\r\n",
        "\r\n",
        "# p(t | s) = p(s | t) * p(t) / p(s)\r\n",
        "\r\n",
        "# p(s) = p(s | t) * p(t) + p(s | e) * p(e) + p(s | a) * p(a)\r\n",
        "\r\n",
        "# p(t | s) = p(s | t) * p(t) / (p(s | t) * p(t) + p(s | e) * p(e) + p(s | a) * p(a))\r\n",
        "\r\n",
        "# p(t | s) = (.5 * .333) / ((.5 * .333) + (.2 * .333) + (.05 * .333))\r\n",
        "\r\n",
        "# p(t | s) = 2/3 or .66667 or 66.667%\r\n",
        "\r\n",
        "# There is a 2/3, .66667, or 66.667% chance that Bill ate turnips."
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tSUqeusUb2nE"
      },
      "source": [
        "# TODO Apply Bayes Theorem Generalized With Different **Priors**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eW3cfJu-cFlZ"
      },
      "source": [
        "Change the prior probabilities of Bill eating these vegetables to a 30% chance he ate turnips, a 10% chance he ate eggplant, and a 60% chance he ate asparagus.\n",
        "\n",
        "Now what is the probability that Bill ate turnips, given that he got sick?\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "L47B0DUG3EEP"
      },
      "source": [
        "## TA Comments"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4gN6AYSM3JgT"
      },
      "source": [
        "There is no answer here!"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "81dbKwHenxnT"
      },
      "source": [
        "# TODO My Report on What I Did and What I Learned"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "C8BeiIern33B"
      },
      "source": [
        "## Fun\n",
        "I had fun because I am learning similar concepts in another class and it was cool to apply them here."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OzOkZ3son-u9"
      },
      "source": [
        "## New\n",
        "The coding problem taught me how to work backwards with the information I know."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mJHvoj6GokaZ"
      },
      "source": [
        "## Meaningful\n",
        "I realized how useful the statistics I have been learning can be in common situations."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CvFLB7eWozwF"
      },
      "source": [
        "## Response\n",
        "I want to understand better how to apply these concepts in the field of computer science."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_RBE03fhpNUG"
      },
      "source": [
        "## Other\n",
        "It was cool seeing the connection between what I am learning in different classes."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Nrk-8EmgW4NP"
      },
      "source": [
        "# TODO What is True?\n",
        "Click on each warranted checkbox to toggle it to True (or back to False). \n",
        "\n",
        "NOTE: *This only works in Colab. If you run it in some other Jupyter notebook client/server environment you may have to change False to True (or vice versa) manually.*\n",
        "\n",
        "This self-assessment is subject to revision by a grader."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qGAjzgVRXR4M"
      },
      "source": [
        "#@markdown ## What is true about what I did?\n",
        "#@markdown ### I had fun.\n",
        "cb00 = True #@param {type:'boolean'}\n",
        "#@markdown ### I learned something new.\n",
        "cb01 = True #@param {type:'boolean'}\n",
        "#@markdown ### I achieved something meaningful, or something I can build upon at a later time.\n",
        "cb02 = True #@param {type:'boolean'}\n",
        "#@markdown ### I watched the linked-to 3Blue1Brown video on Bayes' Theorem, and reported on the experience.\n",
        "cb03 = True #@param {type:'boolean'}\n",
        "#@markdown ## What is true about my report?\n",
        "#@markdown ### I wrote a sufficient number of well-written sentences.\n",
        "cb04 = True #@param {type:'boolean'}\n",
        "#@markdown ### My report is free of mechanical infelicities.\n",
        "cb05 = True #@param {type:'boolean'}\n",
        "#@markdown ### I used Grammarly (or something better described in my report) to check for MIs.\n",
        "cb06 = True #@param {type:'boolean'}\n",
        "#@markdown ### I reported on any connections I found between these problems and something I already know. \n",
        "cb07 = True #@param {type:'boolean'}\n",
        "#@markdown ### I reported who were and what contribution each of my collaborators made.\n",
        "cb08 = False #@param {type:'boolean'}\n",
        "#@markdown ## What is true about my answers?\n",
        "#@markdown ### I got the right answer for the \"Apply Bayes' Theorem\" problem, and my report describes how I know it.\n",
        "cb09 = False #@param {type:'boolean'}\n",
        "#@markdown ### I got the right answer for the \"Compute Four Leaves\" problem, and my report describes how I know it.\n",
        "cb10 = True #@param {type:'boolean'}\n",
        "#@markdown ### I got the right answer for the \"Apply Your Probability Knowledge\" problem, and my report describes how I know it.\n",
        "cb11 = False #@param {type:'boolean'}\n",
        "#@markdown ### I got the right answer for the \"Apply Bayes' Theorem Generalized\" problem, and my report describes how I know it.\n",
        "cb12 = False #@param {type:'boolean'}\n",
        "#@markdown ### I got the right answer for the \"Apply Bayes' Theorem Generalized With Different Priors\" problem, and my report describes how I know it.\n",
        "cb13 = False #@param {type:'boolean'}\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "muxeYgQQKGEU"
      },
      "source": [
        "# For graders"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "c0ldoieNKMTU"
      },
      "source": [
        "#@markdown ---\n",
        "number_of_MIs_found = 0 #@param {type: 'slider', min: 0, max: 5}\n",
        "#@markdown ---\n"
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}