The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern by Keith Devlin

Author:Keith Devlin
Language: eng
Format: mobi, pdf
Publisher: Basic Books
Published: 0101-01-01T00:00:00+00:00

A particularly important concept that Huygens discussed is expectation (or expected gain). Although he declared that none of the work was truly original with him and that he simply built upon the previous work of French mathematicians, in the case of expectation, he is being unduly modest. This concept was at most implicit in earlier work—for instance in Pascal and Fermat’s problem of the unfinished game. By recognizing its significance and making it explicit, Huygens took probability theory a huge step forward.

Expected gain is generally regarded as the correct objective measure (in most cases) of the value of a particular wager to the person who makes it. To compute it, you multiply the probability of each outcome by the amount that will be won (or lost, which you count as a negative gain) and add all the results together. For example, casinos offer even odds for betting on red or black at roulette. Suppose you bet $100 on red. The roulette wheel has 36 slots, numbered from 1 to 36, half of them colored red, half black, and two zeros, colored green. The probability of red coming up is therefore 18/38, that is, 9/19. So your expectation (to the nearest cent) is:

(9/19 × $100) + (10/19 × -$100) = -$100/19 = -$5.26

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