204 lines
5.6 KiB
Plaintext
204 lines
5.6 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "a6732353-51d5-4478-9cf8-5834e57e5a4e",
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"metadata": {},
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"source": [
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"# Chapter 1 Notes"
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]
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},
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{
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"cell_type": "markdown",
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"id": "48d9ec9e-83da-40ca-ae79-3c45f8af137c",
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"metadata": {},
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"source": [
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"## Main Concepts\n",
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"\n",
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"Outcome: A result of a random experiment.\n",
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"\n",
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"Sample Space: The set of all possible outcomes.\n",
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"\n",
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"Event: A subset of the sample space.\n",
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"\n",
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"Inclusion-exclusion principle holds for probability\n",
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"\n",
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"Consider a sample space S. If S is a countable set, this refers to a discrete probability\n",
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"mode\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7ac122be-50b2-423c-b88f-e4b3327b21bd",
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"metadata": {},
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"source": [
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"## Example Problems"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7fb87a35-a470-4d98-935f-80c814e3f95d",
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"metadata": {},
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"source": [
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"Example 1.5 - soln\n",
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"\n",
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"- there are 10 people with white shirts and 8 people with red shirts;\n",
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"- 4 people have black shoes and white shirts\n",
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"- 3 people have black shoes and red shirts\n",
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"- the total number of people with white or red shirts or black shoes is 21\n",
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"\n",
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"Let A be the set of people with white shirts, B be the set of people with red shirts and let C be the set of people with black shoes.\n",
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"\n",
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"\\begin{align*}\n",
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"|A|=10 \\\\\n",
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"|B|=8 \\\\\n",
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"|A \\cap C| = 4 \\\\\n",
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"|B \\cap C| = 3 \\\\\n",
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"|A \\cup B \\cup C| = 21\n",
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"\\end{align*}\n",
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"\n",
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"Now we solve for $|C|$:\n",
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"\n",
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"\\begin{align*}\n",
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"|A| + |B| + |C| - |A \\cap B| - |A \\cap C| - |B \\cap C| + |A \\cap B \\cap C| = 21 \\\\\n",
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"10 + 8 + |C| - 0 - 4 - 3 - 0 = 21 \\\\\n",
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"18 + |C| - 7 = 21 \\\\\n",
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"|C| + 11 = 21 \\\\\n",
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"|C| = 10\n",
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"\\end{align*}\n",
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"\n",
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"$\\therefore$ number of people with black shoes is 10\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "72b40733-531a-48f3-9879-75601684afc2",
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"metadata": {},
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"source": [
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"Example 1.11 - soln\n",
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"\n",
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"Suppose we have the following information:\n",
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"1. There is a 60 percent chance that it will rain today.\n",
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"2. There is a 50 percent chance that it will rain tomorrow.\n",
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"3. There is a 30 percent chance that it does not rain either day.\n",
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"\n",
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"T = rains\n",
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"F = no rain\n",
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"\n",
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"$S = \\{(F, F), (F, T), (T, F), (T, T)\\}$\n",
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"\n",
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"$P((T, F) \\cup (T, T)) = 0.6$\n",
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"\n",
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"$P((F, T) \\cup (T, T)) = 0.5$\n",
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"\n",
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"$P((F, F)) = 0.3$\n",
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"\n",
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"\\begin{align*}\n",
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"P(S) = 1 \\\\\n",
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"P(\\{(F, F)\\} \\cup \\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n",
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"P((F,F)) + P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n",
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"0.3 + P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n",
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"P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 0.7 \\\\\n",
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"P(\\{(F, T)\\} \\cup \\{(T, T)\\}) + P((T, F)) = 0.7 \\\\\n",
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"0.5 + P((T, F)) = 0.7 \\\\\n",
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"P((T, F)) = 0.2 \\\\\n",
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"P(\\{(T, F)\\} \\cup \\{(T, T)\\}) + P((F, T)) = 0.7 \\\\\n",
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"P((F, T)) = 0.1\n",
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"\\end{align*}\n",
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"\n",
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"Find the following probabilities:\n",
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"\n",
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"a. The probability that it will rain today or tomorrow.\n",
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"\n",
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"\\begin{align*}\n",
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"P((T, F) \\cup (F, T) \\cup (T, T)) = 0.7\n",
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"\\end{align*}\n",
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"\n",
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"b. The probability that it will rain today and tomorrow.\n",
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"\n",
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"\\begin{align*}\n",
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"P((T, T)) = 1 - 0.3 - 0.2 - 0.1 = 0.4\n",
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"\\end{align*}\n",
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"\n",
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"c. The probability that it will rain today but not tomorrow.\n",
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"\n",
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"\\begin{align*}\n",
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"P((T, F)) = 0.2\n",
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"\\end{align*}\n",
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"\n",
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"d. The probability that it either will rain today or tomorrow, but not both.\n",
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"\n",
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"\\begin{align*} \n",
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"P(\\{(T, F)\\} \\cup \\{(F, T)\\}) = P((T, F)) + P((F, T)) = 0.2 + 0.1 = 0.3\n",
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"\\end{align*}\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8b5131dd-5ebd-4156-b808-f8df273317fb",
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"metadata": {},
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"source": [
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"Example 1.12 - soln\n",
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"\n",
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"$S = \\{ -1, 0, 1, 2, 3, ... \\}$\n",
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"\n",
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"$\\forall x \\in S, P(x) = \\frac{1}{2^{x + 2}}$\n",
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"\n",
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"What is the probability that I win more than or equal to 1 dollar and less than 4 dollars?\n",
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"\n",
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"\\begin{align*} \n",
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"P({1, 2, 3}) = P(1) + P(2) + P(3) \\\\\n",
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"= 1/8 + 1/16 + 1/32\n",
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"\\end{align*}\n",
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"\n",
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"What is the probability that I win more than 2 dollars?\n",
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"\n",
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"\\begin{align*} \n",
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"\\sum_{i=3}^{\\infty} P(i) = P(3) + P(4) + P(5) + P(6) + ... \\\\\n",
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"= 1/32 + 1/64 + 1/128 + 1/256 + ... \\\\\n",
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"=\\frac{\\frac{1}{32}}{1 - \\frac{1}{2}}\n",
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"=\\frac{1}{16}\n",
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"\\end{align*}"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "69962960-18a6-436b-b9eb-9a6dfae7559a",
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"metadata": {},
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"outputs": [],
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "11163495-a6cc-43b9-bf41-02748b13d210",
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"nbformat": 4,
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"nbformat_minor": 5
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