Table Of ContentUnexpected Expectations: The Curiosities of a
Mathematical Crystal Ball explores how paradoxi-
cal challenges involving mathematical expectation
often necessitate a reexamination of basic prem-
ises. The author takes you through mathematical
paradoxes associated with seemingly straightfor-
ward applications of mathematical expectation
and shows how these unexpected contradictions
may push you to reconsider the legitimacy of the
applications.
Praise for Leonard Wapner’s first book, The Pea and the Sun
The book requires only an understanding of basic
algebraic operations and includes supplemental
“What is presented in this book is maths for its own sake: beautiful, elegant, artistic, mathematical background in chapter appendices.
astonishing...it would surely make a great present for a budding pure mathema- After a history of probability theory, it introduces
tician—and what a present it would be, to give someone their first inkling of the the basic laws of probability as well as the defini-
wonders that lie at the heart of pure mathematics.” tion and applications of mathematical expecta-
tion/expected value (E). The remainder of the text
—Helen Joyce, Plus Magazine, September 2005
Leonard Wapner has taught mathematics at El
covers unexpected results related to mathemati-
Camino College in Torrance, California, since 1973. cal expectation, including:
“This book is sure to intrigue, fascinate, and challenge the mathematically inclined
He received his BA and MAT degrees in mathemat-
ics from the University of California, Los Angeles. reader.” —The Mathematics Teacher, May 2006 • The roles of aversion and risk in rational decision
making
He is the author of The Pea and the Sun: A Math-
ematical Paradox and his writings on mathemat- “Very readable and aimed at non-technical readers … an ideal book for undergradu- • A class of expected value paradoxes referred to
ics education have appeared in The Mathemat- ates or sixth form students.” as envelope problems
ics Teacher (The National Council of Teachers of —Anthony C. Robin, The Mathematical Gazette, November 2006 • Parrondo’s paradox—how negative (losing) ex-
Mathematics) and The AMATYC Review (The pectations can be combined to give a winning
American Mathematical Association of Two-Year result
“One does not need to have a degree in mathematics in order to follow the lively and
Colleges). Len lives in Seal Beach, California, with readable, highly intriguing story of the paradox. Yet the exposition is serious, correct • Problems associated with imperfect recall
his wife Mona and his Labrador Retriever Bailey. and comprehensive, and it presents a detailed proof of the result. The presentation • Non-zero-sum games, such as the game of
is light-hearted, highly entertaining and illustrated with many examples, puzzles, etc. chicken and the prisoner’s dilemma
This book is (already) a classic in an area that needed one.”
• Newcomb’s paradox—a great philosophical par-
—Newsletter of the European Mathematical Society, June 2006 adox 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, math-
ematical expectation generates paradoxes that
frequently leave questions unanswered yet reveal
K13061 interesting surprises. Encouraging you to embrace
the mysteries of mathematics, this book helps you
appreciate the applications of mathematical ex-
pectation, “a statistical crystal ball.”
Mathematics
Unexpected Expectations
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Unexpected Expectations
The Curiosities of a
Mathematical Crystal Ball
Leonard M. Wapner
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2012 by Taylor & Francis Group, LLC
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Version Date: 20120202
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For my mom,
Lea Wapner
For my wife,
Mona Wapner
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Table of Contents
Acknowledgments xi
The Crystal Ball xiii
1. Looking Back 1
Beating the Odds: Girolamo Cardano 3
Vive la France: Blaise Pascal and Pierre de Fermat 6
Going to Press: Christiaan Huygens 8
Law, but No Order: Jacob Bernoulli 10
Three Axioms: Andrei Kolmogorov 13
2. The ABCs of E 19
The Definition of Probability 20
The Laws of Probability 22
Binomial Probabilities 31
The Definition of Expected Value 32
Utility 35
Infinite Series: Some Sum! 37
Appendix 39
3. Doing the Right Thing 41
What Happens in Vegas 41
Is Insurance a Good Bet? 45
Airline Overbooking 47
vii
viii Unexpected Expectations
Composite Sampling 51
Pascal’s Wager 54
Game Theory 56
The St. Petersburg Paradox 63
Stein’s Paradox 64
Appendix 68
4. Aversion Perversion 71
Loss Aversion 72
Ambiguity Aversion 75
Inequity Aversion 78
The Dictator Game 78
The Ultimatum Game 80
The Trust Game 80
Off-Target Subjective Probabilities 82
5. And the Envelope Please! 91
The Classic Envelope Problem: Double or Half 92
The St. Petersburg Envelope Problem 94
The “Powers of Three” Envelope Problem 95
Blackwell’s Bet 97
The Monty Hall Problem 99
Win-Win 104
Appendix 105
6. Parrondo's Paradox: You Can Win for Losing 109
Ratchets 101 109
The Man Engines of the Cornwall Mines 110
Parrondo’s Paradox 112
Reliabilism 114
From Soup to Nuts 115
Parrondo Profits 116
Truels: Survival of the Weakest 117
Going North? Head South! 120
Appendix 123
7. Imperfect Recall 127
The Absentminded Driver 128
Unexpected Lottery Payoffs 131
Sleeping Beauty 134
Applications 137
8. Non-zero-sum Games: The Inadequacy of Individual Rationality 141
Pizza or Pâté 142
The Threat 145
Table of Contents ix
Chicken: The Mamihlapinatapei Experience 146
The Prisoner’s Dilemma 154
The Nash Arbitration Scheme 163
Appendix 167
9. Newcomb's Paradox 169
Dominance vs. Expectation 170
Newcomb + Newcomb = Prisoner’s Dilemma 174
10. Benford's Law 177
Simon Newcomb’s Discovery 178
Benford’s Law 179
What Good Is a Newborn Baby? 184
Appendix 189
Let the Mystery Be! 191
Bibliography 193
Index 199
