792 lines
52 KiB
Plaintext
792 lines
52 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "c58309b2",
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"metadata": {},
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"outputs": [],
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"source": [
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"import os\n",
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"import sys\n",
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"\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"import pandas as pd\n",
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"import seaborn as sns\n",
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"\n",
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"sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
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]
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},
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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 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": "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": "markdown",
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"id": "4bc9a64b",
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"metadata": {},
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"source": [
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"# 1.3.6 Problems. \n",
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"\n",
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"## Problem 1 - skipping\n",
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"\n",
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"## Problem 2 \n",
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"\n",
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"a. \n",
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"\n",
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"\\begin{align*}\n",
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"S = \\{ x \\mid x >= 2, x \\in \\mathbb{N} \\}\n",
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"\\end{align*}\n",
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"\n",
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"b.\n",
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"1 red, 1 blue, 1 white and 1 green\n",
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"\n",
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"\\begin{align*}\n",
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"S = \\{ (R, B), (B, R), (R, W), (W, R), (R, G), (G, R), (B, W), (W, B), (B, G), (G, B), (W, G), (G, W) \\}\n",
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"\\end{align*}\n",
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"\n",
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"c.\n",
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"\n",
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"$S = [0, \\frac{1}{3})$\n",
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"\n",
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"## Problem 3\n",
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"\n",
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"- $A \\cup B \\cup C = S$\n",
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"- $P(A) = \\frac{3}{6}$\n",
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"- $P(B) = \\frac{4}{6}$\n",
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"- $P(A \\cup B) = \\frac{5}{6}$\n",
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"\n",
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"### a. Find $P(A \\cap B)$\n",
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"\n",
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"$P(A \\cap B) = P(A) + P(B) - P(A \\cup B) = \\frac{3}{6} + \\frac{4}{6} - \\frac{5}{6} = \\frac{2}{6}$\n",
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"\n",
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"### b. Do $A$, $B$ and $C$ form a partition of $S$?\n",
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"\n",
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"No. Proof by contridiction:\n",
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"\n",
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"Assume $A$, $B$ and $C$ form a partition of $S$. That would imply $P(S) = P(A) + P(B) + P(C) = 1$.\n",
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"\n",
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"$P(A) = \\frac{3}{6}$ and $P(B) = \\frac{4}{6}$, so $P(A) + P(B) = \\frac{7}{6}$\n",
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"\n",
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"$P(A) + P(B) > P(S)$ therefore contirdiction.\n",
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"\n",
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"$A$, $B$ and $C$ do not form a partition of $S$. $\\blacksquare$\n",
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"\n",
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"### c. Find $P(C \\setminus (A \\cup B))$\n",
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"\n",
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"\\begin{align*}\n",
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"A \\cup B \\cup C = S \\\\\n",
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"A \\cup B \\cup C \\setminus (A \\cup B) = S \\setminus (A \\cup B) \\\\\n",
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"C \\setminus (A \\cup B) = S \\setminus (A \\cup B)\n",
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"\\end{align*}\n",
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"\n",
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"Therefore $P(C \\setminus (A \\cup B)) = P(S \\setminus (A \\cup B))$.\n",
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"\n",
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"Notice $P(S \\setminus (A \\cup B)) = P((A \\cup B)^c)$, so $P(C \\setminus (A \\cup B)) = P((A \\cup B)^c)$\n",
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"\n",
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"So\n",
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"\n",
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"\\begin{align*}\n",
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"P(C - (A \\cup B)) &= P((A \\cup B)^c) \\\\\n",
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"&= P(S) - P(A \\cup B) \\\\\n",
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"&= 1 - \\frac{5}{6} \\\\\n",
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"&= \\frac{1}{6}\n",
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"\\end{align*}\n",
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"\n",
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"### d. If $P(C \\cap (A \\cup B)) = \\frac{5}{12}, P(C) =$ ?\n",
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"\n",
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"$C = (C \\cap (A \\cup B)) \\cup C \\setminus (A \\cup B)$\n",
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"\n",
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"Therefore\n",
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"\n",
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"\\begin{align*}\n",
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"P(C) &= P((C \\cap (A \\cup B)) \\cup C \\setminus (A \\cup B)) \\\\\n",
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"&= P(C \\cap (A \\cup B)) + P(C \\setminus (A \\cup B)) \\\\\n",
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"&= \\frac{5}{12} + \\frac{1}{6} \\\\\n",
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"&= \\frac{7}{12}\n",
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"\\end{align*}\n",
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"\n",
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"## Problem 4.\n",
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"\n",
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"### a. \n",
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"Let $X$ be our sample space for the first row, and $Y$ the second roll. ($X \\cup Y = S$)\n",
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"\n",
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"$X = \\{ 1, 2, 3, 4, 5, 6 \\}$\n",
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"\n",
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"Let $x$ be the first dice row and $y$ the second dice roll. ($x \\in X$ and $y \\in Y$)\n",
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"\n",
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"The probability $x < y$ is $|X| - x$ or $6 - x$.\n",
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"\n",
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"So \n",
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"\n",
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"\\begin{align*}\n",
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"\\sum_{x \\in X} (6 - x) &= \\sum_{x \\in X} 6 - \\sum_{x \\in X} x \\\\\n",
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"&= 36 - (1 + 2 + 3 + 4 + 5 + 6) \\\\\n",
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"&= 15\n",
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"\\end{align*}\n",
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"\n",
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"Therefore $P(A) = \\frac{15}{36}$\n",
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"\n",
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"### b.\n",
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"\n",
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"Number of outcomes where 6 is only first: 1 * 5\n",
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"Number of outcomes where 6 is only second: 1 * 5 \n",
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"Number of outcomes where 6 is both: 1\n",
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"\n",
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"Therefore $P(B) = \\frac{11}{36}$\n",
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"\n",
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"## Problem 5."
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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": "e589ecc1",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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",
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"text/plain": [
|
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"<Figure size 800x500 with 1 Axes>"
|
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]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
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"text": [
|
|
"2.25091334378745\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"xmin = 0\n",
|
|
"xmax = 20\n",
|
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"\n",
|
|
"x = np.linspace(xmin, xmax, 200)\n",
|
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"y = 1 - np.exp((-1 * (x / 5)))\n",
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"\n",
|
|
"# ----------------------------------------------------\n",
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"\n",
|
|
"plt.figure(figsize=(8, 5))\n",
|
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"\n",
|
|
"# 1. Updated the LaTeX label to match your new math\n",
|
|
"plt.plot(x, y, color=\"crimson\", linewidth=2.5, label=r\"$y = e^{-x/5}$\")\n",
|
|
"\n",
|
|
"plt.xlim(xmin, xmax)\n",
|
|
"\n",
|
|
"# 2. Tightened the Y-limits so you can actually see the decay curve\n",
|
|
"plt.ylim(0, 1)\n",
|
|
"\n",
|
|
"plt.legend(fontsize=12, loc=\"upper right\") # Moved to upper right since curve decays downward\n",
|
|
"plt.show()\n",
|
|
"\n",
|
|
"# print((1 - 2.71 ** (-3/5)) * 5)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "00afcbd1",
|
|
"metadata": {},
|
|
"source": [
|
|
"### a.\n",
|
|
"\n",
|
|
"Our sample space $S$ is the time at which the product breaks. The product could break immediatly ($T = 0$) or never break ($T \\to \\infty$)\n",
|
|
"\n",
|
|
"Therefore $S = [0, \\infty)$\n",
|
|
"\n",
|
|
"### b.\n",
|
|
"\n",
|
|
"$P(T \\geq 0) = e^{-\\frac{t}{5}} = 1$\n",
|
|
"\n",
|
|
"This makes sense, as this states that the probability the product doesnt break immediatly is 1. This would have to be the case, otherwise the product was broken to begin with, which is an impossible senario.\n",
|
|
"\n",
|
|
"$\\lim_{t \\to \\infty} P(T \\geq t) = \\lim_{t \\to \\infty} e^{-\\frac{t}{5}} = 0$ (skiping proof, can be easily seen in graph above)\n",
|
|
"\n",
|
|
"This means the as time goes on, the probability the product doesnt break approachs 0. This would imply the product will eventually break given infinite time.\n",
|
|
"\n",
|
|
"### c.\n",
|
|
"\n",
|
|
"$t_1 \\lt t_2 \\implies P(T \\geq t_1) \\geq P(T \\geq t_2)$.\n",
|
|
"\n",
|
|
"Proof\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"t_1 \\leq t_2 \\\\\n",
|
|
"-\\frac{t_1}{5} \\cdot -5 \\geq -\\frac{t_2}{5} \\cdot -5 \\\\\n",
|
|
"-\\frac{t_1}{5} \\geq -\\frac{t_2}{5} \\\\\n",
|
|
"e^{-\\frac{t_1}{5}} \\geq e^{-\\frac{t_2}{5}} \\\\\n",
|
|
"P(T \\geq t_1) \\geq P(T \\geq t_2) \n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"$\\blacksquare$\n",
|
|
"\n",
|
|
"This implies a more time means more probability the product may breakdown which makes intuitive sense.\n",
|
|
"\n",
|
|
"### d. Find the probability that the product breaks down within three years of the purchase time.\n",
|
|
"\n",
|
|
"<!-- \\begin{align*}\n",
|
|
"\\int_{0}^{3} e^{-\\frac{t}{5}} dt\\\\\n",
|
|
"[-5e^{-\\frac{t}{5}} + C]_0^3 \\\\\n",
|
|
"(-5e^{-\\frac{3}{5}} + C) - (-5e^{-\\frac{0}{5}} + C) \\\\\n",
|
|
"-5e^{-\\frac{3}{5}} + C + 5e^{-\\frac{0}{5}} - C \\\\\n",
|
|
"-5e^{-\\frac{3}{5}} + 5 \\\\\n",
|
|
"\\approx 2.251\n",
|
|
"\\end{align*} -->\n",
|
|
"\n",
|
|
"### e. Find the probability that the product breaks down in the second year\n",
|
|
"\n",
|
|
"$P(1 \\leq T \\lt 2)=$?\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(1 \\leq T \\lt 2)=P(T \\geq 1) + P(T \\lt 2)\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"## Problem 6\n",
|
|
"\n",
|
|
"Triangle inequaility:\n",
|
|
"\n",
|
|
"$x + y \\gt z$\n",
|
|
"\n",
|
|
"$y + z \\gt x$\n",
|
|
"\n",
|
|
"$x + z \\gt y$\n",
|
|
"\n",
|
|
"Since $x + y + z = 1$ then:\n",
|
|
"\n",
|
|
"$x \\lt \\frac{1}{2}$\n",
|
|
"\n",
|
|
"$y \\lt \\frac{1}{2}$\n",
|
|
"\n",
|
|
"$z \\lt \\frac{1}{2}$\n",
|
|
"\n",
|
|
"The area of our sample space is the plane where $x + y + z = 1$ in $\\mathbb{R}^3$, which is the triangle with points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$.\n",
|
|
"\n",
|
|
"Area of $S$ is $\\frac{\\sqrt{3}}{2}$ (skipping proof).\n",
|
|
"\n",
|
|
"The triangle $A$ is $\\frac{\\sqrt{3}}{8}$\n",
|
|
"\n",
|
|
"$P(A) = \\frac{|A|}{|S|} = \\frac{\\frac{\\sqrt{3}}{8}}{\\frac{\\sqrt{3}}{2}} = \\frac{\\sqrt{3}}{8} \\cdot \\frac{2}{\\sqrt{3}} = \\frac{1}{4}$\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "4609aba5-dfdc-4500-bec2-cfa2e544cca7",
|
|
"metadata": {},
|
|
"source": [
|
|
"\n",
|
|
"### Example 1.17 - soln\n",
|
|
"\n",
|
|
"I roll a fair die twice and obtain two numbers X1 = result of the first roll and X2 = result\n",
|
|
"of the second roll. Given that I know X1 + X2 = 7, what is the probability that X1 = 4 or\n",
|
|
"X2 = 4?\n",
|
|
"\n",
|
|
"What is our sample space?\n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"D = \\{1,2,3,4,5,6\\} \\\\\n",
|
|
"S = D \\times D\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"P(A|B) = \\frac{P(A \\cap B)}{P(B)}\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"A = \\{(x,y) \\mid x=4 \\lor y = 4 \\} \\\\\n",
|
|
"|A| = 12 \\\\\n",
|
|
"P(A) = \\frac{|A|}{|S|} \\\\\n",
|
|
"= \\frac{12}{36} = \\frac{1}{3}\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"> NOTE my event set A may be overcounting \n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"B = \\{(x,y) \\mid x+y = 7 , x,y \\in \\mathbb{N}\\} \\\\\n",
|
|
"|B| = 6 \\\\\n",
|
|
"P(B) = \\frac{1}{6} \\\\\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"P(A \\cap B) = \\frac{1}{18}\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"B = \\{(x,y) \\mid x+y = 7 , x,y \\in \\mathbb{N}\\} \\\\\n",
|
|
"A = \\{(x,y) \\mid x=4 \\lor y = 4 \\} \\\\\n",
|
|
"P(A|B) = \\frac{P(A \\cap B)}{P(B)} \\\\\n",
|
|
"= \\frac{\\frac{1}{18} }{ \\frac{1}{6} } \\\\\n",
|
|
"= \\frac{6}{18} = \\frac{1}{3}\n",
|
|
"\\end{align*}\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "12c8a0d4",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Example 1.18\n",
|
|
"\n",
|
|
"### a.\n",
|
|
"\n",
|
|
"$S_\\text{conditional} = \\{\\{G, G\\}, \\{G, B\\}\\}$, therefore $\\frac{1}{2}$\n",
|
|
"\n",
|
|
"### b.\n",
|
|
"\n",
|
|
"$S_\\text{conditional} = \\{\\{G, G\\}, \\{G, B\\}, \\{B, G\\}\\}$, therefore $\\frac{1}{3}$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "1305f9d6",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$P(B,C \\mid A) = P(B \\mid A) \\cdot P(C \\mid A,B)$$\n",
|
|
"\n",
|
|
"$$P(A_1 \\cap A_2 \\cap \\dots \\cap A_n \\mid E) = P(A_1 \\mid E) \\cdot P(A_2 \\mid A_1, E) \\cdot P(A_3 \\mid A_1, A_2, E) \\cdots P(A_n \\mid A_1, \\dots, A_{n-1}, E)$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "ca15c14a",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.19\n",
|
|
"\n",
|
|
"In a factory there are 100 units of a certain product, 5 of which are defective. We pick\n",
|
|
"three units from the 100 units at random. What is the probability that none of them are\n",
|
|
"defective?\n",
|
|
"\n",
|
|
"Let $A_1$ be the event where we pick a non defective unit first, $A_2$ on our second, and $A_3$ on our third.\n",
|
|
"\n",
|
|
"We want to find $P(A_1 \\cap A_2 \\cap A_3)$.\n",
|
|
"\n",
|
|
"Note that $P(A_1 \\cap A_2 \\cap A_3) = P(A_1)P(A_2|A_1)P(A_3|A_1,A_2)$\n",
|
|
"\n",
|
|
"$$P(A_1) = \\frac{95}{100}$$\n",
|
|
"\n",
|
|
"$$P(A_2|A_1) = \\frac{94}{99}$$\n",
|
|
"\n",
|
|
"$$P(A_3|A_1,A_2) = \\frac{93}{98}$$\n",
|
|
"\n",
|
|
"\\begin{align*} \n",
|
|
"P(A)P(B|A)P(C|A,B) &= \\frac{95}{100} \\cdot \\frac{94}{99} \\cdot \\frac{93}{98} \\\\\n",
|
|
"\\approx 0.856\n",
|
|
"\\end{align*}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "4790a1dc",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.20\n",
|
|
"\n",
|
|
"$S = [1:11]$\n",
|
|
"\n",
|
|
"Let $A$ be the event $N \\lt 7$, $N \\in S$\n",
|
|
"\n",
|
|
"Let $B$ be the event $N$ is even. Are $A$ and $B$ independent?\n",
|
|
"\n",
|
|
"$$P(A) = \\frac{|A|}{|S|} = \\frac{6}{10} = \\frac{3}{5}$$\n",
|
|
"\n",
|
|
"Note that $|B| = 5$.\n",
|
|
"\n",
|
|
"$$P(A|B) = \\frac{|A \\cap B|}{|B|} = \\frac{3}{5}$$\n",
|
|
"\n",
|
|
"$P(A|B) = P(A) \\therefore A$ and $B$ are independent. \n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "b16a508d",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.21\n",
|
|
"\n",
|
|
"$P(HHHHT) = P(H)P(H)P(H)P(H)P(T) = \\frac{1}{2^5} = \\frac{1}{32}$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "08fe32c4",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.22\n",
|
|
"\n",
|
|
"Number of plane rides over 20 years $= 20 \\cdot 20 = 400$\n",
|
|
"\n",
|
|
"Let $A_i$ be the flight businessman dies where $i \\in [1:401]$\n",
|
|
"\n",
|
|
"We want to find $P(A_1 \\cup A_2 \\cup A_3 \\cup \\dots \\cup A_{400})$\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(A_1 \\cup A_2 \\cup A_3 \\cup \\dots \\cup A_n) &= 1 - (1 - P(A_1))(1 - P(A_2))\\cdots(1 - P(A_{400})) \\\\\n",
|
|
"&= 1 - (1 - \\frac{1}{4\\cdot 10^6})^{400} \\\\\n",
|
|
"\\approx 10^{-4}\n",
|
|
"\\end{align*}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "42589434",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.23\n",
|
|
"\n",
|
|
"a.\n",
|
|
"\n",
|
|
"What is the probability someone makes a shot in a round?\n",
|
|
"\n",
|
|
"Let $A$ be the event player 1 makes the shot, and $B$ the event player 2 makes the shot. \n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(A \\cup B) &= 1 - (1 - P(A))(1 - P(B)) \\\\\n",
|
|
"&= 1 - (1 - p_1)(1 - p_2) \\\\\n",
|
|
"&= p_1 + p_2 - p_1p_2\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"What is the probability nobody makes a shot in a round?\n",
|
|
"\n",
|
|
"$$P((A \\cup B)^c) = 1 - p_1 - p_2 + p_1p_2 = 1 - (1 - p_1)(1 - p_2)$$\n",
|
|
"\n",
|
|
"What is the probability player one wins in the first round?\n",
|
|
"\n",
|
|
"$$P(W_1(1)) = p_1$$\n",
|
|
"\n",
|
|
"What is the probability player one wins in the second round?\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(W_1(2)) = P(R_1(1)) + P(R_1(2)) \\\\\n",
|
|
"= p_1 + P((A \\cup B)^c)p_1 \\\\\n",
|
|
"= p_1 + (1 - (1 - p_1)(1 - p_2))p_1\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"What is the probability player one wins in the third round?\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(W_1(3)) = P(R_1(1)) + P(R_1(2)) + P(R_1(3)) \\\\\n",
|
|
"= p_1 + P((A \\cup B)^c)p_1 + P((A \\cup B)^c)P((A \\cup B)^c)p_1 \\\\\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"What is the probability player one wins on round $n$?\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(W_1(n)) = P(R_1(1)) + P(R_1(2)) + P(R_1(3)) + \\dots + P(R_1(n)) \\\\\n",
|
|
"= p_1 + P((A \\cup B)^c)p_1 + P((A \\cup B)^c)P((A \\cup B)^c)p_1 + \\dots + P((A \\cup B)^c)^np_1 \\\\\n",
|
|
"= p_1 + (1 - p_1 - p_2 + p_1p_2)p_1 + (1 - p_1 - p_2 + p_1p_2)^2p_1 + \\dots + (1 - p_1 - p_2 + p_1p_2)^np_1 \\\\\n",
|
|
"= p_1 (1 + (1 - p_1 - p_2 + p_1p_2) + (1 - p_1 - p_2 + p_1p_2)^2 + \\dots + (1 - p_1 - p_2 + p_1p_2)^n) \n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"Let $x = 1 - p_1 - p_2 + p_1p_2$, substitute for \n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(W_1(n)) = p_1 (1 + x + x^2 + \\dots + x^n) \n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"Notice that $|x| \\lt 1$. Proof\n",
|
|
"\n",
|
|
"$$x = 1 - p_1 - p_2 + p_1p_2 = (1 - p_1)(1 - p_2)$$\n",
|
|
"\n",
|
|
"Since $0<p_1,p_2<1$, we have\n",
|
|
"\n",
|
|
"$0 < (1 - p_1) < 1$ and $0 < (1 - p_2) < 1$ which implies $0 < (1 - p_1)(1 - p_2) < 1$ so $0 < x < 1$ which implies $|x| < 1$\n",
|
|
"\n",
|
|
"Now we consider \n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(W_1) &= p_1 \\cdot \\lim_{n \\to \\infty} (1 + x + x^2 + \\dots + x^n) \\\\\n",
|
|
"&= p_1 \\cdot (\\frac{1}{1 - x}) \\\\\n",
|
|
"&= p_1 \\cdot (\\frac{1}{1 - (1 - p_1 - p_2 + p_1p_2)}) \\\\\n",
|
|
"&= \\frac{p_1}{p_1 + p_2 - p_1p_2} \\\\\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"b.\n",
|
|
"\n",
|
|
"If $P(W_1) = \\frac{1}{2}$, find the values of $p_1$ and $p_2$.\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(W_1) &= \\frac{1}{2} \\\\\n",
|
|
"\\frac{p_1}{p_1 + p_2 - p_1p_2} &= \\frac{1}{2} \\\\\n",
|
|
"2p_1 &= p_1 + p_2 - p_1p_2 \\\\\n",
|
|
"0 &= p_1 + p_2 - p_1p_2 - 2p_1 \\\\\n",
|
|
"0 &= p_2 - p_1p_2 - p_1 \\\\\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"In terms of $p_1$\n",
|
|
"\n",
|
|
"$$p_2 = \\frac{p_1}{1 - p_1}$$\n",
|
|
"\n",
|
|
"In terms of $p_2$\n",
|
|
"\n",
|
|
"$$p_1 = \\frac{p_2}{1 + p_2}$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "7be1e873",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.24\n",
|
|
"\n",
|
|
"Let $A$ be the event where my chosen marble is red.\n",
|
|
"\n",
|
|
"Let $B_1$ be the set of the first bag, $B_2$ the second bag, $B_3$ the third bag.\n",
|
|
"\n",
|
|
"Notice that $B_1, B_2,$ and $B_3$ are disjoint and therefore form a partition of our sample space.\n",
|
|
"\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(A) &= P(A|B_1)P(B_1) + P(A|B_2)P(B_2) + P(A|B_3)P(B_3) \\\\\n",
|
|
"&= \\frac{75}{100} \\cdot \\frac{1}{3} + \\frac{60}{100} \\cdot \\frac{1}{3} + \\frac{45}{100} \\cdot \\frac{1}{3} \\\\\n",
|
|
"&= \\frac{180}{300} = \\frac{6}{10} = 0.6\n",
|
|
"\\end{align*}\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "a3160c4c",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.25\n",
|
|
"\n",
|
|
"Let $A$ be the event we chose a red marble and let $B$ be the event we chose from the first bag.\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(B|A) &= \\frac{P(A|B)P(B)}{P(A)} \\\\\n",
|
|
"&= \\frac{\\frac{75}{100} \\cdot \\frac{1}{3}}{\\frac{6}{10}} \\\\\n",
|
|
"&= \\frac{75}{300} \\cdot \\frac{10}{6} \\\\\n",
|
|
"&= \\frac{750}{1800} = \\frac{75}{180} = \\frac{15}{36} = \\frac{5}{12} \\\\\n",
|
|
"\\end{align*}\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "34782cd8",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Example 1.26\n",
|
|
"\n",
|
|
"Let $A$ be the event we chose a red marble and let $B$ be the event we chose from the first bag.\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(B|A) &= \\frac{P(A|B)P(B)}{P(A)} \\\\\n",
|
|
"\n",
|
|
"\\end{align*}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "086ffc85",
|
|
"metadata": {},
|
|
"source": [
|
|
"### TODO Example 1.27\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"\\end{align*}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "c4ef0787",
|
|
"metadata": {},
|
|
"source": [
|
|
"# TODO 1.4.5 Problems"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "d283ef80",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "roadmap (3.14.5)",
|
|
"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": "3.14.5"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|