Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball

Overview

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 ...

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Overview

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.

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Editorial Reviews

From the Publisher
"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

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Product Details

  • ISBN-13: 9781568817217
  • Publisher: Taylor & Francis
  • Publication date: 6/12/2012
  • Pages: 220
  • Product dimensions: 5.90 (w) x 9.10 (h) x 0.90 (d)

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
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 Mamihlapinatapai Experience
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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