Grinstead-Snell-Probability · Problem 4.1.33

Conditional Probability

Mathematics · mutual-independence-three-events

Problem

Let A1A_1, A2A_2, and A3A_3 be events, and let BiB_i represent either AiA_i or its complement A~i\tilde{A}_i. Then there are eight possible choices for the triple (B1,B2,B3)(B_1, B_2, B_3). Prove that the events A1A_1, A2A_2, A3A_3 are independent if and only if P(B1∩B2∩B3)=P(B1)P(B2)P(B3),P(B_1 \cap B_2 \cap B_3) = P(B_1)P(B_2)P(B_3), for all eight of the possible choices for the triple (B1,B2,B3)(B_1, B_2, B_3).

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