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I'm studying statistics and probability using an introductory textbook and it had this question:

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108.The events “Other” and “Up for reelection in November 2016” are ________

a. mutually exclusive.

b. independent.

c. both mutually exclusive and independent.

d. neither mutually exclusive nor independent.

The answer key says that the correct answer is letter a, but shouldn't it be letter c as the P("Other" $\cap$ "Up for reelection in November 2016") = 0 meaning that the two events are mutually exclusive and that the P("Other"|"Up for reelection in November 2016") = P(Other), therefore also being independent?

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  • $\begingroup$ You can't be mutually exclusive as well as independent. If $A$ and $B$ are m.e., then knowledge of $A$ definitely means that $B$ didn't happen, so they aren't independent. $\endgroup$ – Randall May 21 at 1:29
  • $\begingroup$ If A and B are events and if P(A) = 0, then if two events are mutually exclusive then, P(A AND B) = 0, therefore P(A|B) = P(A), doesn't that mean it's independent too? $\endgroup$ – Joe May 21 at 1:37
  • $\begingroup$ I suppose so, but I don't think impossible events give any deep interpretations. $\endgroup$ – Randall May 21 at 1:40
  • $\begingroup$ I now just read the entire question (sorry). Seems like an odd question. In your case, what is the situation with $P(B \mid A)$? Is this also $P(B)$? $\endgroup$ – Randall May 21 at 1:40
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    $\begingroup$ @Michael I think it's a poorly set-up problem, at least what I can see here. $\endgroup$ – Randall May 21 at 3:06

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