Let A1, A2, and A3 be events, and let Bi represent either Ai or its complement Ai.

Question:

Let A1, A2, and A3 be events, and let Bi represent either Ai or its complement Ai. Then there are eight possible choices for the triple (B1, B2, B3). Prove that the events A1, A2, A3 are independent if and only if
(B1∩B2∩B3) = P (B1) P (B2) P (B3),
For all eight of the possible choices for the triple (B1, B2, B3)