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prob.part8.17_18

# prob.part8.17_18 - 1.1 SIMULATION OF DISCRETE PROBABILITIES...

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1.1. SIMULATION OF DISCRETE PROBABILITIES 9 of the time. A larger number of races would be necessary to have better agreement with the past experience. Therefore we ran the program to simulate 1000 races with our four horses. Although very tired after all these races, they performed in a manner quite consistent with our estimates of their abilities. Acorn won 29.8 percent of the time, Balky 39.4 percent, Chestnut 19.5 percent, and Dolby 11.3 percent of the time. The program GeneralSimulation uses this method to simulate repetitions of an arbitrary experiment with a finite number of outcomes occurring with known probabilities. Historical Remarks Anyone who plays the same chance game over and over is really carrying out a sim- ulation, and in this sense the process of simulation has been going on for centuries. As we have remarked, many of the early problems of probability might well have been suggested by gamblers’ experiences. It is natural for anyone trying to understand probability theory to try simple experiments by tossing coins, rolling dice, and so forth. The naturalist Bu ff on tossed a coin 4040 times, resulting in 2048 heads and 1992 tails. He also estimated the number π by throwing needles on a ruled surface and recording how many times the needles crossed a line (see Section 2.1). The English biologist W. F. R. Weldon 1 recorded 26,306 throws of 12 dice, and the Swiss scientist Rudolf Wolf 2 recorded 100,000 throws of a single die without a computer. Such experiments are very time- consuming and may not accurately represent the chance phenomena being studied.

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