A fair die is rolled twice. Let A be the event that the number on the first die is odd, let B be the event that the number on the second die is odd, and let C be the event that the sum of the two rolls is equal to 7.
a. Show that A and B are independent, A and C are independent, and B and C are independent. This property is known as pairwise independence.
b. Show that A, B, and C are not independent. Conclude that it is possible for a set of events to be pairwise independent but not independent. (In this context, independence is sometimes referred to as mutual independence.)
(a) P(A) = 1?2, P(B) = 1?2, P(A ? B) = 1?4 = P(A)P(B), so A and B are independent.
P(C) = 1?6, P(A ? C) = 1?12 = P(A)P(C), so A and C are independent.
P(B ? C) = 1?12 = P(B)P(C), so B and C are independent.
(b) P(A ? B ? C) = 0 ? P(A)P(B)P(C), so A, B, and C are not independent.
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