assn3-solns - Math 361, Problem set 3 Due 9/20/10 1....

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Math 361, Problem set 3 Due 9/20/10 1. (1.4.21) Suppose a fair 6-sided die is rolled 6 independent times. A match occurs if side i is observed during the i th trial, i =1 , . . . , 6. (a) What is the probability of at least one match during on the 6 rolls. (b) Extend part ( a ) to a fair n -sided die with n independent rolls. Then determine the limit of the probability as n →∞ . Answer: It is much easier to compute the probability that ever roll is a non-match. For any given roll, the probability of a non-match is 5 6 . Since the rolls are independent, the probability that there are no matches is (5 / 6) 6 . Therefore the probability that there is at least one match is 1 - (5 / 6) 6 . Likewise for any roll in the general case the probability of a non-match is n - 1 n , and hence the probability of at least one match is 1 - ± n - 1 n ² n . Since lim n →∞ (1 - 1 /n ) n = e - 1 , in the limit this is 1 - e - 1 . 2. (1.4.32) Hunters A and B shoot at a target; their probabilities of hitting the target are p 1 and p 2 respectively. Assuming independent, can p 1 and p 2 be chosen so that P (0 hits) = P (1 hit) = P (2 hits)?
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assn3-solns - Math 361, Problem set 3 Due 9/20/10 1....

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