Probability, Risk and Reward
Preface
0.1
Motivation
0.2
About the Author
1
Probability Basics
1.1
Introduction
1.2
Chapter Scenario - The Three Card Game
1.3
Terminology
1.4
Example - Rolling a Die
1.5
Simple Events
1.6
The Complement: NOT
1.6.1
The Complement Principle
1.7
Compound Events: AND
1.7.1
Mutually Exclusive Events
1.8
Compound Events: OR
1.8.1
Minding the Overlap
1.9
Compound Events: Conditional Probability
1.9.1
Observations about Conditional Probability
1.10
The Three Card Game Revisited
1.11
Exercises
1.11.1
Exercise - The Monty Hall Problem
1.11.2
Exercise - Twelve-sided Die
2
Tree Diagrams
2.1
Introduction
2.2
Chapter Scenario - Kitty Litter
2.3
Example - Flipping Two Coins
2.4
Terminology Review
2.5
Simulation
2.6
Tree Diagrams
2.7
An Example with Rats
2.7.1
Practice
2.8
Chapter Scenario Revisited - Kitty Litter
2.9
The Urn Model
2.9.1
Practice
2.9.2
An Urn Model with Replacement
2.10
Exercises
2.10.1
Exercise - Three Coins in the Fountain
2.10.2
Exercise - More Coins
2.10.3
Exercise - Class Committee
3
Multiple Events
3.1
Introduction
3.2
Chapter Scenario - The Game of Diff
3.3
Example - Pair of Dice
3.4
Fundamental Counting Principle
3.4.1
Practice
3.4.2
Practice
3.5
Example - Galileo’s Dice Problem
3.6
Chapter Scenario Revisited - The Game of Diff
3.7
Exercises
3.7.1
Exercise - My Die, Your Die
3.7.2
Exercise - Three Dice
3.7.3
Exercise - Dice Games
3.7.4
Exercise - De Morgan’s Laws
4
Interlude - In an Attempt to Kill the Student, the Authors Solve the Same Simple Problem Four Ways (One Bad and Three Good)
4.1
A Problem from the Game of Risk
5
Working with OR Statements
5.1
Introduction
5.2
Chapter Scenario - Paying Taxes
5.3
Class Composition Example
5.3.1
Practice
5.4
Venn Diagrams for Three Events
5.5
Example - Watching Sports
5.6
Revisiting the Chapter Scenario - Paying Taxes
5.7
Exercises
5.7.1
Exercise - We Need More Women Scientists in This World
5.7.2
Exercise - Venn for Four
5.7.3
Exercise - Class Schedules
5.7.4
Exercise - Class Survey
5.7.5
Exercise - Minding the Gap
6
Working with AND and NOT Statements
6.1
Introduction
6.2
Chapter Scenario - The Birthday Problem
6.3
The Chevalier de Mere
6.3.1
Practice
6.3.2
Theorem: The Multiplication Rule
6.3.3
Theorem: Special Case of the Multiplication Rule
6.3.4
Theorem: Complement Rule
6.4
Analysing Game One
6.4.1
Practice
6.5
Analyzing Game Two
6.5.1
Practice
6.6
Revisiting the Chapter Scenario - The Birthday Problem
6.6.1
Practice
6.7
Exercises
6.7.1
Exercise - Birthday Problem continued
6.7.2
Exercise - Slips of Paper
6.7.3
Exercise - Five Coins
6.7.4
Exercise - Random Phone Number
6.7.5
Exercise - Knights of the Round Table
7
Conditional Probability and Bayes’ Theorem
7.1
Introduction
7.2
Chapter Scenario - Drug Testing
7.3
Conditional Probability Basics
7.3.1
Practice
7.4
Example - Kittens
7.5
Conditional Probability and Urn Models
7.5.1
Practice
7.6
Bayes’ Rule
7.7
Example - Defective Fidget Spinners
7.8
Chapter Scenario Revisited - Drug Testing
7.9
Hypothetical 10,000 Table
7.10
Exercises
7.10.1
Exercise - Three Kids
7.10.2
Exercise - Condemned Man
7.10.3
Exercise - Liberal and Conservative Urns
8
Mathematical Expectation
8.1
Introduction
8.2
Chapter Scenario - Deal or No Deal
8.3
Introducing Mathematical Expectation
8.4
Example - Win Six
8.5
Example - Chuck-a-Luck
8.6
Chapter Scenario Revisited - Deal or No Deal
8.6.1
Practice
8.6.2
Simulating Deal or No Deal
8.6.3
Practice
8.6.4
Reflection
8.7
Deal or No Deal Extended
8.8
Exercises
8.8.1
Exercise - Slot Machine
8.8.2
Exercise - Design Your Own Slot Machine
8.8.3
Exercise - Fair Chuck-a-Luck
8.8.4
Exercise - Chuck-a-Luck Big Prize
8.8.5
Exercise - St. Petersburg Paradox
8.8.6
Exercise - Squared Dots
8.8.7
Exercise - Raising the Stakes
8.8.8
Exercise - Deal or No Deal Revisited
8.8.9
Exercise - Deal or No Deal On Your Own
9
Measures of Variation
9.1
Introduction
9.2
Chapter Scenario - Roulette Fun Bus
9.3
Measuring Variability with Variance and Standard Deviation
9.3.1
Definition of Variance
9.3.2
Definition of Standard Deviation
9.4
Example - Roulette Even Money Bet
9.5
Example - Roulette Single Number Bet
9.6
Example - Deal or No Deal
9.7
Chapter Scenario Revisited - Roulette Fun Bus
9.7.1
Simulating the Bus
9.7.2
Simulating the Bus
9.7.3
Comparing the and Buses
10
Permutations
10.1
Introduction
10.2
Chapter Scenario - GATC
10.3
Permutations
10.4
Fundamental Counting Principle Principle
10.5
Permutation Formula
10.6
Example - The Line-up
10.7
Example - The MISSISSIPPI Problem
10.8
Exercises
10.9
Exercises - Scrabble
10.9.1
Exercise - The Name Game
10.9.2
Exercise - License Plates
10.9.3
Exercise - Facts about Permutions
11
Combinations
11.1
Introduction
11.2
Chapter Scenario - Three Counting Problems and an Algebra Problem
11.3
Example - From Class Line-up to Class Committee
11.4
Combination Formula
11.5
Chapter Scenario Revisited - Three Counting Problems and an Algebra Problem
11.5.1
Problem 1 - Birth Orderings
11.5.2
Problem 2 - Block-walking
11.5.3
Problem 3 - Team Captains
11.5.4
Problem 4 - Binomial Expansion
11.6
Exercises
11.6.1
Exercise - Alphabet
11.6.2
Exercise - Facts about Combinations
12
Case Study - Block-walking, Tree Diagrams, and the Quincunx
12.1
Chapter Scenario - The Quincunx
12.2
The Block-walking Encyclopedia
12.3
Chapter Scenario Revisited - The Quincunx
12.4
Simulation
12.4.1
Exercise - Two-stage Quincunx
12.4.2
Exercise - Plinko
13
Case Study - The Problem of the Points
13.1
Chapter Scenario - The BART Series
13.2
A Simpler Case - A Three-Two Split
13.3
Chapter Scenario Revisited - The BART Series
13.3.1
Exercise - A Three:One Split
13.3.2
Exercise - A Three:Zero Split
14
The Binomial Distribution
14.1
Introduction
14.2
Chapter Scenario - Pounding Pennies
14.3
The Binomial Distribution
14.4
The Binomial Distribution in R
14.5
Example - Throwing Darts
14.5.1
Individual Probabilities
14.5.2
Cumulative Probabilities
14.5.3
Inverse Probabilities
14.6
Example - Tennis First Serve Percentage
14.7
Example - Set, Spike, Kill
14.8
Chapter Scenario Revisited - Pounding Pennies
14.9
Exercises
14.9.1
Exercises - Birth Orderings
14.9.2
Exercise - Slot Machine
14.9.3
Exercise - Jasmine Recommends
14.9.4
Exercise - Cancer Indicence
14.9.5
Exercise - Famous Places
14.9.6
Exercise - Stealing Dogs
14.9.7
Exercise - Please Read this Book!
14.9.8
Exercise - The Animal Pictionary
14.9.9
Exercise - Not that Funny!
14.9.10
Exercise - Beat the Dean
14.9.11
Exercise - Perro No Habla Espanol
14.9.12
Exercise - The Kitty Genovese Case
14.9.13
Exercise - Sweeping Three
14.9.14
Exercise - Stop Browsing Pinterest During Class
14.9.15
Exercise - To Sweat or Not to Sweat
14.9.16
Exercise - I Lift Things Up, I Set Things Down
14.9.17
Exercise - Liking Probability
14.9.18
Exercise - Swimming in the Ocean
14.9.19
Exercise - Biking the Double Black Diamond
14.9.20
Exercise - Sloth-lore
14.9.21
Exercise - DIY Binomial
15
Normal Distribution
15.1
Introduction
15.2
Chapter Scenario - Wanna be a Fighter Pilot
15.3
The Normal Probability Distribution
15.4
The Normal Distribution in R
15.5
Example - Normal Blood Pressure
15.5.1
Finding SBP Probabilities with
pdist
15.5.2
Finding SBP Percentiles with
qdist
15.6
The Standard Normal
\(Z \sim N(0,1)\)
15.7
Example - Tides
15.8
The 68-95-99.7 Rule
15.9
Exercises
15.9.1
Exercise - Exercise, Exercise
15.9.2
Exercise - Hello, Doggie!
15.9.3
Exercise - Is ICE Nice?
15.9.4
Exercise - Friday is Skiday
15.9.5
Exercise - Ballroom Burn
15.9.6
Exercise - Don’t Search Pinterest During Probability Class
15.9.7
Exercise - Normal Blood Pressure
15.9.8
Exercise - Ping Pong with the Dean
15.9.9
Exercise - Should I Take This to Antique Road Show?
15.9.10
Exercise - Should be Have Universal Health Care?
15.9.11
Exercise - Movie Run Times
15.9.12
Exercise - The Sloth Olympics
15.9.13
Exercise (Group Project) - Analyzing Grading Systems
16
Sums of Random Variables
16.1
Introduction
16.2
Chapter Scenario - Comparing Heights
16.3
Example - Wheels of Fortune
16.3.1
Practice - Preconceptions
16.3.2
Simulating the Games
16.4
Expectation for the Sum of Two Random Variables
16.5
Variance and Standard Deviation for the Sum of Two Random Variables
16.6
Summing Independent Identically Distributed Random Variables
16.7
The Roots of the Normal
16.7.1
Simulating the Toss of Two Dice
16.8
The Central Limit Theorem
16.8.1
The Central Limit for Sums
16.8.2
The Central Limit for Sample Means
16.9
Conclusion
16.10
Adding Normals
16.11
Example - Tips on the Weekend
16.12
Chapter Scenario Revisited - Comparing Heights
16.13
Exercises
16.13.1
Exercise (Group Project) - Zombie Apocalypse
16.13.2
Exercise - Dungeons and Dragons Dice
16.13.3
Exercise - Difference of Wheels
16.13.4
Exercise - Comparing Blood Pressure
17
The Geometric Distribution
17.1
Introduction
17.2
Chapter Scenario - Lovin’ the Airport
17.3
Example - Waiting for Lucky Number 7
17.3.1
The Geometric Distribution
17.3.2
The Geometric Random Variable in R
17.4
Exercises
17.4.1
Exercise - The Geometric Dice Game
17.4.2
Exercise - NBA Halfcourt Shot
17.4.3
Exercise - The Chevalier de Mere Revisited
18
The Hypergeometric Distribution
18.1
Introduction
18.2
The Hypergeometric Random Variable
18.2.1
Example - A Class Controversy
18.2.2
The Hypergeometric Distribution
18.2.3
The Hypergeometric Random Variable in R
18.3
Exercises
19
The Poisson Distribution
19.1
Introduction
19.2
The Poisson Random Variable
19.2.1
The Poisson Distribution
19.2.2
The Poisson Random Variable in R
19.2.3
Modeling Number of Earthquakes with the Poisson
19.3
Exercises
19.3.1
Exercise - Counting Earthquakes
19.3.2
Exercise - Growth Mindset and Math Errors
20
Multinomial Distribution
20.1
Chapter Scenario - 3D Ant Walking
20.2
Chapter Scenario Revisited - 3D Ant Walking
20.2.1
A Simpler Example
20.2.2
The Original Problem
20.2.3
General 3D Lattice Situation
20.3
Example - Hair Color
21
The Uniform Distribution
21.1
Introduction
21.2
Chapter Scenario - Lovin’ the Airport
21.3
The Uniform Distribution
21.3.1
The Uniform Probability Density Function
21.3.2
The Uniform Random Variable in R
21.3.3
Example - The UNIF(0,1) Random Variable
21.3.4
Example - Catch a Flight
21.4
Chapter Scenario Revisited - Lovin’ the Airport
21.5
Exercises
21.5.1
Exercise - Transforming UNIF(0,1)
21.5.2
Exercise - Showing Up for Class
22
The Exponential Distribution
22.1
Introduction
22.2
The Exponential Distribution
22.2.1
The Exponential Probability Density Function
22.2.2
The Exponential Random Variable in R
22.2.3
Example - Soccer Wait Times
22.2.4
Example - The Computer Help Desk
22.3
Exercises
22.3.1
Exercise - Waiting on that Earthquake
22.3.2
Exercise - NHL Hockey Goals
23
Monte Carlo Simulation
23.1
Introduction
23.2
Chapter Scenario - Random Darts
23.3
About Random Number Generators
23.4
Chapter Scenario Revisited - Random Darts
23.5
Example - The Holey Cube
23.6
Simulating The Cube
23.7
Exercises
23.7.1
Exercise - City Grid
23.7.2
Exercise - A Big Bead
24
Case Study - Roulette
24.1
Chapter Scenario - Conservative or Playa?
24.2
The Rules of Roulette
24.3
Analyzing the Even-Money Bets
24.4
Roulette Simulation
24.5
Analyzing the Single Number Bet
24.6
Exercises
24.6.1
Exercise - Four and Five Number Bets
24.6.2
Exercise - Simulating a Roulette Bet
24.6.3
Exercise - The Rule of 36
24.6.4
Exercise - European Roulette
25
Case Study - Craps
25.1
Chapter Scenario - Getting Six the Hard Way
25.2
Single Roll Craps Bets
25.3
The Craps Passline Bet Explained
25.4
Chapter Scenario Revisited - Getting Six the Hard Way
25.5
Exercises
25.5.1
Exercise - Single Roll Bet
25.5.2
Exercise - Multi-roll Bet
25.5.3
Exercise - The Don’t Pass Bet
25.5.4
Exercise - Free Odds
26
Case Study - Poker
26.1
Introduction
26.1.1
The Royal Flush
26.1.2
Straight Flush
26.1.3
Four of a Kind
26.1.4
Full House
26.1.5
Flush
26.1.6
Straight
26.1.7
Three of a Kind
26.1.8
Two Pair
26.1.9
One Pair
26.1.10
High Card
26.1.11
Summary Table
26.2
Exercises
26.2.1
Exercise - Straight Flush
26.2.2
Exercise - Four of a Kind
26.2.3
Exercise - One Pair
26.2.4
Exercise - High Card
26.2.5
Exercise - Spot the Error
27
Case Study - Keno
27.1
Analyzing a One-spot Keno Ticket
27.2
Analyzing a Two-spot Keno Ticket
27.3
Exercises
27.3.1
Exercise - Analyzing a Three-spot and Four-spot Keno Tickets
27.3.2
Exercise - Changing Perspectives
27.3.3
Exercise (Group Project) - Complete Analysis of Keno
28
Case Study - Powerball Lottery
28.1
Introduction
28.2
Ways to Win Powerball
28.2.1
Winning Powerball with Five White
28.2.2
Winning Powerball with Four White + Powerball
28.2.3
Winning Powerball with Four White
28.2.4
Winning Powerball with Three White + Powerball
28.2.5
Winning Powerball with Three White
28.2.6
Winning Powerball with Two White + Powerball
28.2.7
Winning Powerball with One White + Powerball
28.2.8
Winning Powerball with Powerball Only
28.3
The Expected Value of Powerball
28.4
Looking for a Winner
28.5
Exercises
28.5.1
Exercise - Ways to Win Powerball
28.5.2
Exercise - Can I Get a Winner?
28.5.3
Exercise - Today: 522 Million Dollar Jackpot
28.5.4
Exercise - Think Positive Expectation
28.5.5
Exercise - Thinking about Dependencies
29
Case Study - BINGO
29.1
Chapter Scenario - BINGO on the Fun Bus
29.2
Bingo Basics
29.3
Winning Bingo with m Numbers Called
29.4
Chapter Scenario Revisited - BINGO on the Fun Bus
29.5
Exercises
29.5.1
Exercise - Unique BINGO Cards
29.5.2
Exercise - Win with Seven
30
Genetics
30.1
Appendix 1: Genetics Examples
30.1.1
Genetics Terminology
30.1.2
Genetics Basics
30.1.3
Genetics Simulation
References
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Probability, Risk, and Reward
References