By Leonard M. Wapner

ISBN-10: 1568817215

ISBN-13: 9781568817217

Unexpected expectancies: The Curiosities of a Mathematical Crystal Ball explores how paradoxical demanding situations related to mathematical expectation usually necessitate a reexamination of simple premises. the writer takes you thru mathematical paradoxes linked to possible effortless functions of mathematical expectation and exhibits how those unforeseen contradictions could push you to reassess the legitimacy of the applications.

The publication calls for merely an realizing of uncomplicated algebraic operations and comprises supplemental mathematical historical past in bankruptcy appendices. After a historical past of likelihood conception, it introduces the elemental legislation of likelihood in addition to the definition and functions of mathematical expectation/expected price (E). the rest of the textual content covers unforeseen effects relating to mathematical expectation, including:
• the jobs of aversion and threat in rational determination making
• a category of anticipated price paradoxes often called envelope problems
• Parrondo’s paradox—how adverse (losing) expectancies might be mixed to provide a successful consequence
• difficulties linked to imperfect recall
• Non-zero-sum video games, akin to the sport of bird and the prisoner’s dilemma
• Newcomb’s paradox—a nice philosophical paradox of loose will
• Benford’s legislation and its use in machine layout and fraud detection


While invaluable in components as various as video game idea, quantum mechanics, and forensic technological know-how, mathematical expectation generates paradoxes that regularly depart questions unanswered but exhibit attention-grabbing surprises. Encouraging you to embody the mysteries of arithmetic, this ebook is helping you relish the functions of mathematical expectation, "a statistical crystal ball."

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Extra resources for Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball

Sample text

Rolling a sum of 2 can be done in only one way, as (1,1); therefore, the probability of rolling a sum of 2 is 1/36. Rolling a sum of 3 is twice as likely because rolling a 3 can be done in two ways: (1,2) and (2,1). The most likely sum to roll is 7. The six ways of doing so are depicted in S along the diagonal running from the lower left to the upper right. The probability of rolling a 7 is therefore 6/36 = 1/6. 1. With these probabilities in mind, the law of large numbers can be illustrated. 2 give the relative frequencies of sums from an experiment in which a pair of dice was rolled 20 and 100 times.

If each simple event is portrayed in the form (outcome of the first die, outcome of the second die), then the sample space is given by  (1,1), (1, 2),   (2,1), (2, 2),   (3,1), (3, 2),  S=  (4 ,1), (4 , 2),   (5,1), (5, 2),   (6,1), (6, 2), (1, 3), (1, 4 ), (1, 5), (2, 3), (2,, 4 ), (2, 5), (3, 3), (3, 4 ), (3, 5), (4 , 3), (4 , 4 ), (4 , 5), (5, 3), (5, 4 ), (5,, 5), (6, 3), (6, 4 ), (6, 5), (1, 6),   (2, 6),   (3, 6),   . (4 , 6),   (5, 6),   (6 , 6 )   Chapter 2. 1.

Fortunately he had on a parachute. Unfortunately the parachute didn’t open. Fortunately there was a haystack below. Unfortunately there was a pitchfork in the haystack. Fortunately he missed the pitchfork. Unfortunately he missed the haystack! In order to establish a formula for calculating conditional probabilities, we consider once again the simple act of randomly drawing one card from a standard deck. The probability of drawing the king of hearts is p(K♥) = 1/52. We draw the card and before we have a chance to look at it, we are told reliably that the card is a king.

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Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball by Leonard M. Wapner

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