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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, BC N 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(AN 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.)

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, BC N 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(AN 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.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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(ANB) = 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 Ak be the event that all the digits are k or smaller. What is the probability of Ar, 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.
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(ANB) = 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 Ak be the event that all the digits are k or smaller. What is the probability of Ar, 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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