Let A and B be events with P(A)=0.3 and P(A U B)=0.7.
a. For what value of P(B) will A and B be mutually exclusive?
b. For what value of P(B) will A and B be independent?
(a) A and ? are mutually exclusive if P(A ?B) = 0, or equivalently, if
= P(A) + P(B). So if P(B) = – P(A) = 0.7 – 0.3 = 0.4, then A and ? are mutually exclusive.
(b) A and ? are independent if P(A ??) = P(A)P(B). Now = P(A) + P(B) – P(A ?B).
So A and B are independent if = P(A) + P(B) – P(A)P(B), that is, if 0.7 = 0.3 + P(B) – 0.3P(B). This equation is satisfied if P(B) = 4/7.
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