Contents

Preface

1 Computing Probabilities: Right Ways and Wrong Ways

THE PROBABILIST

THE PROBABILIST’S TOYS AND LANGUAGE

THE PROBABILIST’S RULE BOOK

INDEPENDENCE, AIRPLANES, AND RUSSIAN PEASANTS

CONDITIONAL PROBABILITY, SWEDISH TV, AND BRITISH COURTS

LIAR, LIAR

TOTAL PROBABILITY, USED CARS, AND TENNIS MATCHES

COMBINATORICS, PASTRAMI, AND POETRY

THE VON TRAPPS AND THE BINOMIAL DISTRIBUTION

FINAL WORD

2 Surprising Probabilities: When Intuition Struggles

BOYS, GIRLS, ACES, AND COLORED CARDS

GOATS AND GLOATS

HAPPY BIRTHDAY

TYPICAL ATYPICALITIES

STRATEGIES, SHOPPING, AND SPAGHETTI WESTERNS

THE BRITISH SNOB AND I

FINAL WORD

3 Tiny Probabilities: Why Are They So Hard to Escape?

PROBABLE IMPROBABILITIES

SADDAM AND I

TAKING TINY RISKS

A MILLION-TO-ONE SHOT, DOC, MILLION TO ONE!

MONSIEUR POISSON AND THE MYSTERIOUS NUMBER 37

CLUMPS IN SPACE

FINAL WORD

4 Backward Probabilities: The Reverend Bayes to Our Rescue

DRIVING MISS DAISY

BAYES, BALLS, AND BOYS (AND GIRLS)

BAYES AND MY GREEN CARD

OBJECTION YOUR HONOR

FINAL WORD

5 Beyond Probabilities: What to Expect

GREAT EXPECTATIONS

GOOD THINGS COME TO THOSE WHO WAIT

EXPECT THE UNEXPECTED

SIZE MATTERS (AND LENGTH, AND AGE)

DEVIANT BEHAVIOR

FINAL WORD

6 Inevitable Probabilities: Two Fascinating Mathematical Results

ALEA IACTA EST, OVER AND OVER

EVEN-STEVEN? THE LAW MISUNDERSTOOD

COIN TOSSES AND FREEWAY CONGESTION

LET’S GET SERIOUS

BELLS AND BREAD

HOW A TORONTO QUINCUNX CHANGED MY LIFE

FINAL WORD

7 Gambling Probabilities: Why Donald Trump Is Richer than You

FRENCH LETTERS

ROULETTE: A CLASSY WAY TO WASTE YOUR MONEY

CRAPS: NOT SO DICEY AFTER ALL

BLACKJACK: MONEY FOR MNEMONICS

MATH FOR LOSERS

WIN MONEY AND LOSE FRIENDS

FINAL WORD

8 Guessing Probabilities: Enter the Statisticians

LIES, DAMNED LIES, AND BEAUTIFUL LIES?

4 OUT OF 10 LIKE THE PRESIDENT 19 TIMES OUT OF 20

POLLS GONE WILD

THE LAWSUIT AND THE LURKER

FOOTBALL PLAYERS AND GEYSER ERUPTIONS

SNOOPING IN THE ABBOT’S GARDEN

FINAL WORD

9 Faking Probabilities: Computer Simulation

MAHOGANY DICE AND MODULAR ARITHMETIC

RANDOM AND NOT-SO-RANDOM DIGITS

NUMBER ONE IS NUMBER ONE

IS RANDOM REALLY RANDOM?

FINAL WORD

Index

THE WILEY BICENTENNIAL-KNOWLEDGE FOR GENERATIONS

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Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

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Library of Congress Cataloging-in-Publication Data is available.

ISBN-13: 978-0-470-04001-0

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Preface

This book is about those little numbers that we just cannot escape. Try to remember the last day you didn’t hear at least something about probabilities, chance, odds, randomness, risk, or uncertainty. I bet it’s been a while. In this book, I will tell you about the mathematics of such things and how it can be used to better understand the world around you. It is not a textbook though. It does not have little colored boxes with definition or theorems, nor does it contain sections with exercises for you to solve. My main purpose is to entertain you, but it is inevitable that you will also learn a thing or two. There are even a few exercises for you, but they are so subtly presented that you might not even notice until you have actually solved them.

The spousal thanks is always more than a formality. I thank Aλμήvη for putting up with irregular work hours and everything else that comes with writing a book, but also for help with Greek words and for reminding me of some of my old travel stories that you will find in the book. I am deeply grateful to Professor Olle Häggström at Chalmers University of Technology in Göteborg, Sweden. He has read the entire manuscript, and his comments are always insightful, accurate, and clinically free from unnecessary politeness. If you find something in this book that strikes you as particularly silly, chances are that Mr. Häggström has already pointed it out to me but that I decided to keep it for spite. I have also received helpful comments from John Haigh at the University of Sussex, Steve Quigley at Wiley, and from an anonymous referee. Thanks also to Kris Parrish and Susanne Steitz at Wiley, to Sheree Van Vreede at Sheree Van Vreede Publications Services for excellent copyediting, and to Amy Hendrickson at Texnology Inc. for promptly and patiently answering my LaTeX questions.

A large portion of this book was written during the tumultuous Fall of 2005. Our move from Houston to New Orleans in early August turned out to be a masterpiece of bad timing as Hurricane Katrina hit three weeks later. We evacuated to Houston, and when Katrina’s sister Rita approached, we took refuge in the deserts of West Texas and New Mexico. Sandstorms are so much more pleasant than hurricanes! However, it was also nice to return to New Orleans in January 2006; the city is still beautiful, and its chargrilled oysters are unsurpassed. I am grateful to many people who housed us and helped us in various ways during the Fall and by doing so had direct or indirect impact on this book. Special thanks to Kathy Ensor & Co. at the Department of Statistics at Rice University in Houston and to Tom English & Co. at the College of the Mainland in Texas City for providing me with office space. Finally, thanks to Professor Peter Jagers at Chalmers University of Technology, who as my Ph.D. thesis advisor once in a distant past wisely guided me through my first serious encounters with probabilities, those little numbers that rule our lives.

PETER OLOFSSON
www.peterolofsson.com

New Orleans,2006

1

Computing Probabilities: Right Ways and Wrong Ways

THE PROBABILIST

Whether you like it or not, probabilities rule your life. If you have ever tried to make a living as a gambler, you are painfully aware of this, but even those of us with more mundane life stories are constantly affected by these little numbers. Some examples from daily life where probability calculations are involved are the determination of insurance premiums, the introduction of new medications on the market, opinion polls, weather forecasts, and DNA evidence in courts. Probabilities also rule who you are. Did daddy pass you the X or the Y chromosome? Did you inherit grandma’s big nose? And on a more profound level, quantum physicists teach us that everything is governed by the laws of probability. They toss around terms like the Schrödinger wave equation and Heisenberg’s uncertainy principle, which are much too difficult for most of us to understand, but one thing they do mean is that the fundamental laws of physics can only be stated in terms of probabilities. And the fact that Newton’s deterministic laws of physics are still useful can also be attributed to results from the theory of probabilities. Meanwhile, in everyday life, many of us use probabilities in our language and say things like “I’m 99% certain” or “There is a one-in-a-million chance” or, when something unusual happens, ask the rhetorical question “What are the odds?”

Some of us make a living from probabilities, by developing new theory and finding new applications, by teaching others how to use them, and occasionally by writing books about them. We call ourselves probabilists. In the universities, you find us in mathematics and statistics departments; there are no departments of probability. The terms “mathematician” and “statistician” are much more well known than “probabilist,” and we are a little bit of both but we don’t always like to admit it. If I introduce myself as a mathematician at a cocktail party, people wish they could walk away. If I introduce myself as a statistician, they do. If I introduce myself as a probabilist…well, most actually still walk away. They get upset that somebody who sounds like the Swedish Chef from the Muppet Show tries to impress them with difficult words. But some stay and give me the opportunity to tell them some of the things I will now tell you about.

Let us be etymologists for a while and start with the word itself, probability. The Latin roots are probare, which means to test, to prove, or to approve, and habilis, which means apt, skillful, able. The word “probable” was originally used in the sense “worthy of approval,” and its connection to randomness came later when it came to mean “likely” or “reasonable.” In my native Swedish, the word for probable is “sannolik,” which literally means “truthlike” as does the German word “wahrscheinlich.” The word “probability” still has room for nuances in the English language, and Merriam-Webster’s online dictionary lists four slightly different meanings. To us a probability is a number used to describe how likely something is to occur, and probability (without indefinite article) is the study of probabilities.

Probabilities are used in situations that involve randomness. Many clever people have thought about and debated what randomness really is, and we could get into a long philosophical discussion that could fill the rest of the book. Let’s not. The French mathematician Pierre-Simon Laplace (1749–1827) put it nicely: “Probability is composed partly of our ignorance, partly of our knowledge.” Inspired by Monsieur Laplace, let us agree that you can use probabilities whenever you are faced with uncertainty. You could:

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!