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Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball

Leonard M. Wapner
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Bollman
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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 ( 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
Infinite Series: Some Sum!

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

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

Parrondo’s Paradox: You Can Win for Losing
Ratchets 101
The Man Engines of the Cornwall Mines
Parrondo’s Paradox
From Soup to Nuts
Parrondo Profits
Truels—Survival of the Weakest
Going North? Head South!

Imperfect Recall
The Absentminded Driver
Unexpected Lottery Payoffs
Sleeping Beauty

Non-zero-sum Games: The Inadequacy of Individual Rationality
Pizza or Pâté
The Threat
Chicken: The Mamihlapinatapai Experience
The Prisoner’s Dilemma
The Nash Arbitration Scheme

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?

Let the Mystery Be!