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."

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