# Probability and Statistics Made Interesting

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Based on years of experience teaching online and in-person, * Probability and Statistics Made Interesting* integrates the author’s classroom-tested and proven methods and examples to make probability and statistics interesting while still covering the necessary subject matter for a one semester, elementary course.

**Probability and Statistics Made Interesting**

- provides examples that introduce a variety of statistical concepts to motivate students. For example: What is randomness? What’s the point of taking a random sample? How do casino games demonstrate the law of averages?
- integrates chapters on the normal curve, conditional probability, and a chapter on “What’s Luck Got to do with it” that integrates challenging examples.
- examines both large and small sample tests for proportions and means, including Fisher’s Exact test, as well as a test to see if someone is accident prone or just unlucky.
- integrates technology such as smartphones, calculators, Excel, and Google Sheets.
- features discussion topics and problems at the end of each chapter

**NEW - Examples from the Second Edition (available soon) **

Simpson’s Paradox – Two drugs that treat mental illness – Which is better?

Berkson’s Paradox – Comparing restaurants: Does good bacon predict good eggs?

Inspection Paradox – Are average class sizes larger than average?

Random Lightning Strikes – Tragedy in Lafayette Park across from the White House

Predicting the Price of Ethereum (incorrectly)

Excess Deaths and the Covid Vaccine in England – Once again, correlation doesn’t mean causation

So-called psychics use inside information to fool their victims

The Prosecutor’s Fallacy – Fingerprint evidence may not be as strong as it seems

**From the First Edition **

Probability and gambling games – Can you win playing roulette?

Lotteries – If your chance is 1 in 303 million of winning the Mega Millions lottery jackpot, how come there are winners?

Stereotyping – Is the person described as shy more likely to be a farmer or a librarian? From “Thinking, Fast or Slow” by Daniel Kahneman

Clinical trials – Why is a clinical trial better than an observational study?

Zener Card Test for Psychic Powers – What can go wrong?

“Synchronicity” vs randomness – Which is it?

Acknowledgments

Preface

**CHAPTER 1 Random Sampling and Describing Data **

Spreadsheets

Randomness

Random Sampling

Using Excel to Select a Random Sample

Public Opinion Polls

The Inspection Paradox —Estimating Average Class Size

Train Arrival Times

Face-to-Face Surveys

Erroneous Data

Sample Size Matters

Summarizing the Results of a Random Sample

The Mean and the Median

Skewed Data

The Mode

Excel

Variation

Range

Standard Deviation

Excel

Shape

Excel

Bell-Shaped Data

**CHAPTER 2 Probability **

Measuring Randomness

A Brief History of Probability

The Basic Ingredients

Random Variation

The Law of Averages

The Gambler’s Fallacy

Cherry Picking

Casino Games

Craps

Roulette

The Addition Rule for Expectation-Multiple Bets

Pie Throwing

The Double-Down Strategy and the Karaoke Strategy

Bold Play

Watch the Wheel

Dr. Jarecki

That’s Entertainment

Blackjack and Simulations

Computer Simulations

The Kelly Betting System

Poker

The Mega Millions Lottery

Buy Many Tickets

Buy Every Ticket

Splitting the Jackpot Prize

The Station

A Tragic Strategy

The Impossible Lottery

Raffles

The Chance of Being Struck by Lightning

Lightning Strikes at Random

The Bad Luck Lottery

The Chance of Being Killed by a Hornet, Wasp, or Bee Sting

Probability Models

**CHAPTER 3 Conditional Probability **

Cards

Comparing Categorical Variables – Smoking versus Blood Pressure

Asthma and Diet

Astrology

Does Stepping on a Crack in the Sidewalk Really Bring You Bad Luck?

Multiplication Rule for Conditional Probability

Bayes’ Rule—Test For a Rare Disease

Stereotyping—Librarian or Farmer?

Raccoons and Rabies

A Guessing Game

Simpson’s Paradox

Student Admissions

The Nowhere Man

**CHAPTER 4 Correlation and Regression **

Relationships

Square Footage versus Selling Price

Smoking and Pregnancy

Predicting the Weather

The Correlation Coefficient

Random Variation, Sometimes Known as Random Error

Correlation versus Causation

The Regression Line

Using Excel for Regression

The Coefficient of Determination: r

Berkson’s Paradox – Bacon and Eggs

Some Things aren’t Linear

Fuel Efficiency

Two Half-Lives Do Not Make a Whole Life

Good Things (Like Regression) Don’t Last Forever

Predicting the Price of Ethereum (suggested by Katrina Uy)

Song of the Sirens

Excess Deaths and the Covid Vaccine in England

**CHAPTER 5 The Normal Distribution **

The Normal Curve and Probability Histograms

The Empirical Rule

Finding Normal Probabilities

Smooth Curves and Not-So-Precise Measurements

Using Z

Tails of Z

Other normal curves

Th is One’s a Turkey

And the Winner is . . .

“Reverse” Normal Computations

**CHAPTER 6 Statistical Inference—Margin of Error and Confidence Intervals **

Balls in a Box

Good Vibrations

Standard Error

The Central Limit Theorem

The Die is Cast . . . Numerous Times

How Large is Large?

Margin of Error

Confidence Intervals for Proportions

A Public Opinion Poll

Anthropology

The Cable Guy

Determining the Sample Size for Precision and Accuracy

Confidence Intervals for Means

Waiting in Line

Fire Control

**CHAPTER 7 Hypothesis Tests and p-Values **

Yes–No Decisions (Sorry, no “Maybe”)

A Market Research Problem

p-value

Significance Level

Excuse the Formality

Yet Another Guessing Game

Madam Zola

Psychics Sometimes Use Inside Information to Gain Confidence

Chance Needs Opportunity

More Zener Card Guessing

Cherry Picking

Meaningful and Meaningless

False Significance

The Bonferroni Correction

Predicting Stock Market Crashes

The Hamster Tout

Small Sample Test for a Proportion—the Binomial Test

Internet Advertising

Hypothesis Tests for Means

Small Sample Test for Means

Kicking the Tires (statistically speaking)

Two-Tailed Tests

Cloud Seeding and Rainfall

**CHAPTER 8 Tests Comparing Proportions and Means **

If You’re Happy and You Know It, Clap Your Hands

Clinical Trial to Test a New Vaccine

Blind and Double Blind

Efficacy

Relationships

Smoking and Blood Pressure

Asthma and Diet

She (He, They) Loves Me, She (He, They) Loves Me Not

The Experiment

Comparing Means for Two Independent Samples – Knee Replacement

The Not-So-Dumb Waiter

Small Sample Tests for Comparing Two Means—the Two-Sample t-test

Automobile Maintenance Costs—Gas Cars vs. Electric Cars

**CHAPTER 9 The Hypergeometric, Binomial, and Geometric Distributions With Special Bonus – the Fisher Exact Test **

The Multiplication Rule Revisited

Faces

Balls in Boxes

Getting Your Ducks in a Row

Combinations

The Mega Millions Lottery Revisited

The Impossible Lottery Revisited

Poker Hands

The Hypergeometric Distribution

Quality Control

Fisher’s Exact Test

Jury Selection

Again with the Coin Tossing

The Binomial Distribution

Internet Advertising Revisited

The Binomial Formula

Dogs and Fleas

Back to Coin Tossing

Betting on 7 in Craps Revisited

The Prosecutor’s Fallacy

Mean and Standard Deviation of the Binomial Distribution

Normal Approximation

The Hypergeometric Distribution Compared with the Binomial Distribution

The Geometric Distribution

The Assembly Line

Mean of the Geometric Distribution

**CHAPTER 10 What’s Luck Got to Do With It? **

What is Luck?

Luck Is All There Is

Roulette

Luck in Everyday Life

Prediction Games—What’s Luck Got to Do with It?

p-values

Multiple Comparisons

The Binomial Model

The Mega Millions Lottery Revisited

Lucky Winners

A Lucky Winner Needs Unlucky Losers

Accident Proneness—Give Chance a Chance

Run Over by a Bus . . . Twice

Using Data to Estimate a Probability

Bus Accidents in New York City

Finding the p-Value—Binomial and Poisson Distributions

Multiple Comparisons

Multiple Tickets

The Importance of Opportunity

A Statistician, an Insurance Agent, and Carl Jung Walk Into a Bar

On the Lookout

Statistics to the Rescue

*Answers to Problems
Index*

**Michael Orkin**

Dr. Michael Orkin is a professor, consultant, and author. He’s been an invited speaker at numerous events, including conferences, college graduations, rotary clubs, and technical venues such as Google Tech Talks (“Decision Making and Chance,” September 2006). He’s the author of several previous books, including “*Can You Win? The Real Odds for Casino Gambling, Sports Betting, and Lotteries*,” and “*What are the Odds? Chance in Everyday Life*.” Orkin has published numerous research articles in probability and statistics, including “*Games of Chance and Games of Skill*,” which appeared in a recent edition of *Chance* magazine. Dr. Orkin has a B.A. in Mathematics and Ph.D. in Statistics, both from the University of California at Berkeley. He is currently Professor of Mathematics at Berkeley City College in Berkeley, California and is Professor of Statistics, Emeritus, at California State University, East Bay.

**Follow Dr. Michael Orkin**

**Probability and Statistics Made Interestin**g is really what the students need. May it be used everywhere. It’s really all they need.**Dr. Robert Pisani, Statistics Dept., University of California, Berkeley (ret.)****Co-Author of ****Statistics**** by Freedman, Pisani, and Purves**

Based on years of experience teaching online and in-person, * Probability and Statistics Made Interesting* integrates the author’s classroom-tested and proven methods and examples to make probability and statistics interesting while still covering the necessary subject matter for a one semester, elementary course.

**Probability and Statistics Made Interesting**

- provides examples that introduce a variety of statistical concepts to motivate students. For example: What is randomness? What’s the point of taking a random sample? How do casino games demonstrate the law of averages?
- integrates chapters on the normal curve, conditional probability, and a chapter on “What’s Luck Got to do with it” that integrates challenging examples.
- examines both large and small sample tests for proportions and means, including Fisher’s Exact test, as well as a test to see if someone is accident prone or just unlucky.
- integrates technology such as smartphones, calculators, Excel, and Google Sheets.
- features discussion topics and problems at the end of each chapter

**NEW - Examples from the Second Edition (available soon) **

Simpson’s Paradox – Two drugs that treat mental illness – Which is better?

Berkson’s Paradox – Comparing restaurants: Does good bacon predict good eggs?

Inspection Paradox – Are average class sizes larger than average?

Random Lightning Strikes – Tragedy in Lafayette Park across from the White House

Predicting the Price of Ethereum (incorrectly)

Excess Deaths and the Covid Vaccine in England – Once again, correlation doesn’t mean causation

So-called psychics use inside information to fool their victims

The Prosecutor’s Fallacy – Fingerprint evidence may not be as strong as it seems

**From the First Edition **

Probability and gambling games – Can you win playing roulette?

Lotteries – If your chance is 1 in 303 million of winning the Mega Millions lottery jackpot, how come there are winners?

Stereotyping – Is the person described as shy more likely to be a farmer or a librarian? From “Thinking, Fast or Slow” by Daniel Kahneman

Clinical trials – Why is a clinical trial better than an observational study?

Zener Card Test for Psychic Powers – What can go wrong?

“Synchronicity” vs randomness – Which is it?

Acknowledgments

Preface

**CHAPTER 1 Random Sampling and Describing Data **

Spreadsheets

Randomness

Random Sampling

Using Excel to Select a Random Sample

Public Opinion Polls

The Inspection Paradox —Estimating Average Class Size

Train Arrival Times

Face-to-Face Surveys

Erroneous Data

Sample Size Matters

Summarizing the Results of a Random Sample

The Mean and the Median

Skewed Data

The Mode

Excel

Variation

Range

Standard Deviation

Excel

Shape

Excel

Bell-Shaped Data

**CHAPTER 2 Probability **

Measuring Randomness

A Brief History of Probability

The Basic Ingredients

Random Variation

The Law of Averages

The Gambler’s Fallacy

Cherry Picking

Casino Games

Craps

Roulette

The Addition Rule for Expectation-Multiple Bets

Pie Throwing

The Double-Down Strategy and the Karaoke Strategy

Bold Play

Watch the Wheel

Dr. Jarecki

That’s Entertainment

Blackjack and Simulations

Computer Simulations

The Kelly Betting System

Poker

The Mega Millions Lottery

Buy Many Tickets

Buy Every Ticket

Splitting the Jackpot Prize

The Station

A Tragic Strategy

The Impossible Lottery

Raffles

The Chance of Being Struck by Lightning

Lightning Strikes at Random

The Bad Luck Lottery

The Chance of Being Killed by a Hornet, Wasp, or Bee Sting

Probability Models

**CHAPTER 3 Conditional Probability **

Cards

Comparing Categorical Variables – Smoking versus Blood Pressure

Asthma and Diet

Astrology

Does Stepping on a Crack in the Sidewalk Really Bring You Bad Luck?

Multiplication Rule for Conditional Probability

Bayes’ Rule—Test For a Rare Disease

Stereotyping—Librarian or Farmer?

Raccoons and Rabies

A Guessing Game

Simpson’s Paradox

Student Admissions

The Nowhere Man

**CHAPTER 4 Correlation and Regression **

Relationships

Square Footage versus Selling Price

Smoking and Pregnancy

Predicting the Weather

The Correlation Coefficient

Random Variation, Sometimes Known as Random Error

Correlation versus Causation

The Regression Line

Using Excel for Regression

The Coefficient of Determination: r

Berkson’s Paradox – Bacon and Eggs

Some Things aren’t Linear

Fuel Efficiency

Two Half-Lives Do Not Make a Whole Life

Good Things (Like Regression) Don’t Last Forever

Predicting the Price of Ethereum (suggested by Katrina Uy)

Song of the Sirens

Excess Deaths and the Covid Vaccine in England

**CHAPTER 5 The Normal Distribution **

The Normal Curve and Probability Histograms

The Empirical Rule

Finding Normal Probabilities

Smooth Curves and Not-So-Precise Measurements

Using Z

Tails of Z

Other normal curves

Th is One’s a Turkey

And the Winner is . . .

“Reverse” Normal Computations

**CHAPTER 6 Statistical Inference—Margin of Error and Confidence Intervals **

Balls in a Box

Good Vibrations

Standard Error

The Central Limit Theorem

The Die is Cast . . . Numerous Times

How Large is Large?

Margin of Error

Confidence Intervals for Proportions

A Public Opinion Poll

Anthropology

The Cable Guy

Determining the Sample Size for Precision and Accuracy

Confidence Intervals for Means

Waiting in Line

Fire Control

**CHAPTER 7 Hypothesis Tests and p-Values **

Yes–No Decisions (Sorry, no “Maybe”)

A Market Research Problem

p-value

Significance Level

Excuse the Formality

Yet Another Guessing Game

Madam Zola

Psychics Sometimes Use Inside Information to Gain Confidence

Chance Needs Opportunity

More Zener Card Guessing

Cherry Picking

Meaningful and Meaningless

False Significance

The Bonferroni Correction

Predicting Stock Market Crashes

The Hamster Tout

Small Sample Test for a Proportion—the Binomial Test

Internet Advertising

Hypothesis Tests for Means

Small Sample Test for Means

Kicking the Tires (statistically speaking)

Two-Tailed Tests

Cloud Seeding and Rainfall

**CHAPTER 8 Tests Comparing Proportions and Means **

If You’re Happy and You Know It, Clap Your Hands

Clinical Trial to Test a New Vaccine

Blind and Double Blind

Efficacy

Relationships

Smoking and Blood Pressure

Asthma and Diet

She (He, They) Loves Me, She (He, They) Loves Me Not

The Experiment

Comparing Means for Two Independent Samples – Knee Replacement

The Not-So-Dumb Waiter

Small Sample Tests for Comparing Two Means—the Two-Sample t-test

Automobile Maintenance Costs—Gas Cars vs. Electric Cars

**CHAPTER 9 The Hypergeometric, Binomial, and Geometric Distributions With Special Bonus – the Fisher Exact Test **

The Multiplication Rule Revisited

Faces

Balls in Boxes

Getting Your Ducks in a Row

Combinations

The Mega Millions Lottery Revisited

The Impossible Lottery Revisited

Poker Hands

The Hypergeometric Distribution

Quality Control

Fisher’s Exact Test

Jury Selection

Again with the Coin Tossing

The Binomial Distribution

Internet Advertising Revisited

The Binomial Formula

Dogs and Fleas

Back to Coin Tossing

Betting on 7 in Craps Revisited

The Prosecutor’s Fallacy

Mean and Standard Deviation of the Binomial Distribution

Normal Approximation

The Hypergeometric Distribution Compared with the Binomial Distribution

The Geometric Distribution

The Assembly Line

Mean of the Geometric Distribution

**CHAPTER 10 What’s Luck Got to Do With It? **

What is Luck?

Luck Is All There Is

Roulette

Luck in Everyday Life

Prediction Games—What’s Luck Got to Do with It?

p-values

Multiple Comparisons

The Binomial Model

The Mega Millions Lottery Revisited

Lucky Winners

A Lucky Winner Needs Unlucky Losers

Accident Proneness—Give Chance a Chance

Run Over by a Bus . . . Twice

Using Data to Estimate a Probability

Bus Accidents in New York City

Finding the p-Value—Binomial and Poisson Distributions

Multiple Comparisons

Multiple Tickets

The Importance of Opportunity

A Statistician, an Insurance Agent, and Carl Jung Walk Into a Bar

On the Lookout

Statistics to the Rescue

*Answers to Problems
Index*

**Michael Orkin**

Dr. Michael Orkin is a professor, consultant, and author. He’s been an invited speaker at numerous events, including conferences, college graduations, rotary clubs, and technical venues such as Google Tech Talks (“Decision Making and Chance,” September 2006). He’s the author of several previous books, including “*Can You Win? The Real Odds for Casino Gambling, Sports Betting, and Lotteries*,” and “*What are the Odds? Chance in Everyday Life*.” Orkin has published numerous research articles in probability and statistics, including “*Games of Chance and Games of Skill*,” which appeared in a recent edition of *Chance* magazine. Dr. Orkin has a B.A. in Mathematics and Ph.D. in Statistics, both from the University of California at Berkeley. He is currently Professor of Mathematics at Berkeley City College in Berkeley, California and is Professor of Statistics, Emeritus, at California State University, East Bay.

**Follow Dr. Michael Orkin**

**Probability and Statistics Made Interestin**g is really what the students need. May it be used everywhere. It’s really all they need.**Dr. Robert Pisani, Statistics Dept., University of California, Berkeley (ret.)****Co-Author of ****Statistics**** by Freedman, Pisani, and Purves**