26. Let A and B be two events. Suppose P{A} > 0 and P{B} > 0. If A and B are ind

Question

26. Let A and B be two events. Suppose P{A} > 0 and P{B} > 0. If A and B are independent, can they also be mutually exclusive? Why or why not?

27. Let the sample space S = {aaa, bbb, ccc, abc, bca, cba, acb, bac, cab}, and let the probability that each element of S occurs is 1/9. (That is, all outcomes are equally likely.) Let the event Ai be the event that the ith place in the triple is occupied by a. That is, A1 is the event that the outcome is aaa, abc, or acb, since the first place is occupied by a in these outcomes. Also, the event A1 A2 would mean that both the first and second places in the triple are occupied by a. Calculate P{A1}, P{A2}, P{A3}, and P{A1 A2}, P{A1 A3}, P{A2 A3}, and P{A1A2A3}. Use the definitions of pairwise independence and mutual independence to determine whether the events A1, A2, and A3 are pairwise independent or mutually independent. In the writeup of your answer, fully justify your arguments (use the definitions!).

Explanation / Answer

26.

We can prove they can be mutually exclusive by a counter example.

Take for example a coin toss and a die roll. They are independent, as they don't influence each other's results.

Also, they are mutually exclusive, as there is no result in a coin toss that are the same with that of a die roll.

Thus, in this example, we saw that two events can both be indepedent and mutually exclusive.

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