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Publisher:

Chapman & Hall/CRC

Publication Date:

2012

Number of Pages:

204

Format:

Hardcover

Price:

39.95

ISBN:

9781568817217

Category:

General

[Reviewed by , on ]

Mark Bollman

07/30/2012

The notion of expected value is, of course, fundamental material in an introductory statistics course, and finds a place in many “liberal arts mathematics” courses. Leonard Wapner’s *Unexpected Expectations* begins with the basic notion of expectation, but then uses it as a unifying theme for a large collection of mathematical ideas that at first glance, might seem far removed from elementary probability and simple arithmetic.

After a quick exploration of the history of probability, the big idea of expected value is introduced, and then it’s off to a wide range of applications, many of them standard (gambling, insurance, and airline booking among them), but some of them very ambitious for a book aimed at a general audience. These latter examples include Benford’s Law, Parrondo’s Paradox, and some elementary results from game theory. The mathematical arguments are present for those who wish to explore them, but the thrust of the book is to illustrate the myriad of applications of the simple formula for expected value, independent of the mathematical justification that underlies them. At this, *Unexpected Expectations* is a success.

In *The Pea and the Sun*, Wapner took the Banach-Tarski Paradox and brought it to a mainstream audience at a very accessible level. While his subject in this book is not nearly as mathematically complicated, he has done the same thing for mathematical expectation, and the result is an excellent contribution to popular mathematics writing.

Mark Bollman (mbollman@albion.edu) is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

**The Crystal Ball**

** **

**Looking Back **Beating the Odds: Girolamo Cardano

Vive la France: Blaise Pascal and Pierre de Fermat

Going to Press: Christiaan Huygens

Law, but No Order: Jacob Bernoulli

Three Axioms: Andrei Kolmogorov

**The ABCs of ****E **

The Definition of Probability

The Laws of Probability

Binomial Probabilities

The Definition of Expected Value

Utility

Infinite Series: Some Sum!

Appendix

**Doing the Right Thing **What Happens in Vegas

Is Insurance a Good Bet?

Airline Overbooking

Composite Sampling

Pascal’s Wager

Game Theory

The St. Petersburg Paradox

Stein’s Paradox

Appendix

**Aversion Perversion **Loss Aversion

Ambiguity Aversion

Inequity Aversion

The Dictator Game

The Ultimatum Game

The Trust Game

Off-Target Subjective Probabilities

**And the Envelope Please! **The Classic Envelope Problem: Double or Half

The St. Petersburg Envelope Problem

The "Powers of Three" Envelope Problem

Blackwell’s Bet

The Monty Hall Problem

Win-Win

Appendix

**Parrondo’s Paradox: You ***Can* **Win for Losing **Ratchets 101

The Man Engines of the Cornwall Mines

Parrondo’s Paradox

Reliabilism

From Soup to Nuts

Parrondo Profits

Truels—Survival of the Weakest

Going North? Head South!

Appendix

**Imperfect Recall **The Absentminded Driver

Unexpected Lottery Payoffs

Sleeping Beauty

Applications

**Non-zero-sum Games: The Inadequacy of Individual Rationality **Pizza or Pâté

The Threat

Chicken: The

The Prisoner’s Dilemma

The Nash Arbitration Scheme

Appendix

**Newcomb’s Paradox**

Dominance vs. Expectation

Newcomb + Newcomb = Prisoner’s Dilemma

**Benford’s Law **Simon Newcomb’s Discovery

Benford’s Law

What Good Is a Newborn Baby?

Appendix

**Let the Mystery Be! **

**Bibliography **

**Index**

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