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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 paradoxhow 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 paradoxa 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.
|Publisher:||Taylor & Francis|
|Product dimensions:||5.90(w) x 9.10(h) x 0.90(d)|
Table of Contents
The Crystal Ball
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
The Definition of Expected Value
Infinite Series: Some Sum!
Doing the Right Thing
What Happens in Vegas
Is Insurance a Good Bet?
The St. Petersburg Paradox
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
The Monty Hall Problem
Parrondo’s Paradox: You Can Win for Losing
The Man Engines of the Cornwall Mines
From Soup to Nuts
TruelsSurvival of the Weakest
Going North? Head South!
The Absentminded Driver
Unexpected Lottery Payoffs
Non-zero-sum Games: The Inadequacy of Individual Rationality
Pizza or Pâté
Chicken: The Mamihlapinatapai Experience
The Prisoner’s Dilemma
The Nash Arbitration Scheme
Dominance vs. Expectation
Newcomb + Newcomb = Prisoner’s Dilemma
Simon Newcomb’s Discovery
What Good Is a Newborn Baby?
Let the Mystery Be!