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Unformatted text preview: 1.1. SIMULATION OF DISCRETE PROBABILITIES 13 3 In the early 1600s, Galileo was asked to explain the fact that, although the number of triples of integers from 1 to 6 with sum 9 is the same as the number of such triples with sum 10, when three dice are rolled, a 9 seemed to come up less often than a 10—supposedly in the experience of gamblers. (a) Write a program to simulate the roll of three dice a large number of times and keep track of the proportion of times that the sum is 9 and the proportion of times it is 10. (b) Can you conclude from your simulations that the gamblers were correct? 4 In raquetball, a player continues to serve as long as she is winning; a point is scored only when a player is serving and wins the volley. The first player to win 21 points wins the game. Assume that you serve first and have a probability .6 of winning a volley when you serve and probability .5 when your opponent serves. Estimate, by simulation, the probability that you will win a game. 5 Consider the bet that all three dice will turn up sixes at least once in n rolls of three dice. Calculate f ( n ), the probability of at least one triple-six when three dice are rolled n times. Determine the smallest value of n necessary for a favorable bet that a triple-six will occur when three dice are rolled n times. (DeMoivre would say it should be about 216 log2 = 149 . 7 and so would answer 150—see Exercise 1.2.17. Do you agree with him?) 6 In Las Vegas, a roulette wheel has 38 slots numbered 0, 00, 1, 2, . .. , 36. TheIn Las Vegas, a roulette wheel has 38 slots numbered 0, 00, 1, 2, ....
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- Spring '09