A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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solution for c2

C1. Let A and B be two events with P(A) = 0.8 and P(B) = 0.4. The following questions concern the value of P(An B), the probability that both A and B occur. (a) Prove the upper bound P(AN B) < 0.4. (b) Prove that this upper bound can be achieved, by giving an example of a sample space N, a probability measure P and events A, B CN such that P(A) = 0.8 and P(B) = 0.4, with equality P(AN B) = 0.4 in the upper bound. (c) Give a lower bound for P(AN B) – that is, prove that P(A n B) > something – and show that this lower bound can be achieved. (Try to work out the correct bound, even if you can't formally prove it.) C2. A “random digit" is a number chosen at random from {0,1, ...,9}, each with equal probability. A statistician choses n random digits. (a) For k = 0, 1,... , 9, let A be the event that all the digits are k or smaller. What is the probability of A, as a function of k and n? (b) Let Br be the event that the largest digit chosen is equal to k. What is the probability of B? Justify your answer carefully.
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Transcribed Image Text:C1. Let A and B be two events with P(A) = 0.8 and P(B) = 0.4. The following questions concern the value of P(An B), the probability that both A and B occur. (a) Prove the upper bound P(AN B) < 0.4. (b) Prove that this upper bound can be achieved, by giving an example of a sample space N, a probability measure P and events A, B CN such that P(A) = 0.8 and P(B) = 0.4, with equality P(AN B) = 0.4 in the upper bound. (c) Give a lower bound for P(AN B) – that is, prove that P(A n B) > something – and show that this lower bound can be achieved. (Try to work out the correct bound, even if you can't formally prove it.) C2. A “random digit" is a number chosen at random from {0,1, ...,9}, each with equal probability. A statistician choses n random digits. (a) For k = 0, 1,... , 9, let A be the event that all the digits are k or smaller. What is the probability of A, as a function of k and n? (b) Let Br be the event that the largest digit chosen is equal to k. What is the probability of B? Justify your answer carefully.
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