Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball (Hardback) book cover

Unexpected Expectations

The Curiosities of a Mathematical Crystal Ball

By Leonard M. Wapner

© 2012 – A K Peters/CRC Press

220 pages

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Hardback: 9781568817217
pub: 2012-06-04
eBook (VitalSource) : 9781439867679
pub: 2012-06-04
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About the Book

Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball explores how paradoxical challenges involving mathematical expectation often necessitate a reexamination of basic premises. The author takes you through mathematical paradoxes associated with seemingly straightforward applications of mathematical expectation and shows how these unexpected contradictions may push you to reconsider the legitimacy of the applications.

The book requires only an understanding of basic algebraic operations and includes supplemental mathematical background in chapter appendices. After a history of probability theory, it introduces the basic laws of probability as well as the definition and applications of mathematical expectation/expected value (E). The remainder of the text covers unexpected results related to mathematical expectation, including:

  • The roles of aversion and risk in rational decision making
  • A class of expected value paradoxes referred to as envelope problems
  • Parrondo’s paradox—how negative (losing) expectations can be combined to give a winning result
  • Problems associated with imperfect recall
  • Non-zero-sum games, such as the game of chicken and the prisoner’s dilemma
  • Newcomb’s paradox—a great philosophical paradox of free will
  • Benford’s law and its use in computer design and fraud detection

While useful in areas as diverse as game theory, quantum mechanics, and forensic science, mathematical expectation generates paradoxes that frequently leave questions unanswered yet reveal interesting surprises. Encouraging you to embrace the mysteries of mathematics, this book helps you appreciate the applications of mathematical expectation, "a statistical crystal ball."

Listen to an interview with the author on NewBooksinMath.com.


"Apart from its general interest, this book contains much material which would be suitable for presentation to students at many levels of achievement. When full details are not included, there are adequate references to the literature, and there is a good balance between mathematics and qualitative description."

—Peter Giblin, The Mathematical Gazette, March 2014

"the book is highly recommended as Leonard Wapner has a great writing style that both comforts and challenges the reader in the world of confusing paradoxes. Most of the content was new to me, yet it was enjoyable and helped ‘un-solidify’ my belief in mathematical expectations. Get the book and read it if you enjoy being puzzled and discovering new things about what you thought you understood!"

—MathNEXUS, March 2014

"Expectation is an extension of the idea of average value, and is a basic tool of probability theory that underlies both the gaming and insurance industries. Unexpected Expectations is a fascinating look at some of the counterintuitive aspects of this apparently simple concept."

—Jim Stein, NewBooksinMath.com, May 2013

"Every reader—unless they are encyclopedic consumers of all things related to mathematical expectation both in technical journals and the popular press—will find illuminating discussions of paradoxical probabilities that are new to them."

—Andrew James Simoson, Mathematical Reviews, January 2013

"… 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. … an excellent contribution to popular mathematics writing."

—Mark Bollman, MAA Reviews, July 2012

Table of Contents

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!



Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Geometry / General
MATHEMATICS / Recreations & Games
MATHEMATICS / Probability & Statistics / Bayesian Analysis