By Leonard M. Wapner
Unexpected expectancies: The Curiosities of a Mathematical Crystal Ball explores how paradoxical demanding situations regarding mathematical expectation frequently necessitate a reexamination of easy premises. the writer takes you thru mathematical paradoxes linked to likely straight forward functions of mathematical expectation and exhibits how those unforeseen contradictions could push you to reassess the legitimacy of the applications.
The publication calls for in simple terms an knowing of simple algebraic operations and contains supplemental mathematical history in bankruptcy appendices. After a historical past of likelihood idea, it introduces the fundamental 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 on the topic of mathematical expectation, including:
• the jobs of aversion and danger in rational selection making
• a category of anticipated price paradoxes often called envelope problems
• Parrondo’s paradox—how detrimental (losing) expectancies could be mixed to provide a successful outcome
• difficulties linked to imperfect recall
• Non-zero-sum video games, similar to the sport of fowl and the prisoner’s dilemma
• Newcomb’s paradox—a nice philosophical paradox of unfastened will
• Benford’s legislation and its use in machine layout and fraud detection
While helpful in parts as various as online game conception, quantum mechanics, and forensic technology, mathematical expectation generates paradoxes that often depart questions unanswered but demonstrate attention-grabbing surprises. Encouraging you to embody the mysteries of arithmetic, this e-book is helping you get pleasure from the functions of mathematical expectation, "a statistical crystal ball."
take heed to an interview with the writer on NewBooksinMath.com.
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Extra info for Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball
The potential payoff associated with the “×” option is mind-numbingly large, albeit highly unlikely. 7 × 10113), and we tend to appreciate this in terms of its intrinsic worth, or utility, rather than its actual value. So, we compare the expected utilities of the two options, rather than the expected payoffs. To simplify the discussion we use the common (base 10) logarithm of a payoff as a measure of its utility. (The common logarithm of a positive real number is the power to which 10 must be raised to yield the number.
Up or down? If, say, 1,000 such tacks are tossed and 160 land point up, we could invoke Bernoulli’s law of large numbers and assume the true probability is approximately equal to the experimental probability of 160/1000 = 4/25 or 16 percent. A more reliable estimate could be achieved by tossing a larger number of tacks and recording the results. In practice empirical probabilities are used as estimates of true probabilities because the true (theoretical) probabilities are not generally accessible.
Doing the Right Thing 43 is the case for all casino games; otherwise, the casino would lose money and go out of business. For Keno the house advantage can be as large as 27 percent, and for slot machines it varies between 4 percent and 8 percent.
Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball by Leonard M. Wapner