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Unformatted text preview: 2.1. SIMULATION OF CONTINUOUS PROBABILITIES 51 Length of Number of Number of Estimate Experimenter needle casts crossings for π Wolf, 1850 .8 5000 2532 3.1596 Smith, 1855 .6 3204 1218.5 3.1553 De Morgan, c.1860 1.0 600 382.5 3.137 Fox, 1864 .75 1030 489 3.1595 Lazzerini, 1901 .83 3408 1808 3.1415929 Reina, 1925 .5419 2520 869 3.1795 Table 2.1: Buffon needle experiments to estimate π . presented a number of mortality tables and used them to compute, for each age group, the expected remaining lifetime. From his table he observed: the expected remaining lifetime of an infant of one year is 33 years, while that of a man of 21 years is also approximately 33 years. Thus, a father who is not yet 21 can hope to live longer than his one year old son, but if the father is 40, the odds are already 3 to 2 that his son will outlive him. 6 Buffon wanted to show that not all probability calculations rely only on algebra, but that some rely on geometrical calculations. One such problem was his famous “needle problem” as discussed in this chapter. 7 In his original formulation, Buffon describes a game in which two gamblers drop a loaf of French bread on a wide-board floor and bet on whether or not the loaf falls across a crack in the floor. Buffon asked: what length L should the bread loaf be, relative to the width W of the floorboards, so that the game is fair. He found the correct answer (floorboards, so that the game is fair....
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- Spring '09