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."

Reviews

… 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 excelleibution to popular mathematics writing.

—Mark Bollman, MAA Reviews, July 2012

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

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...