1. The problem statement, all variables and given/known data Consider three events A1, A2, and A3, and let pi = P(Ai), for i = 1, 2, 3. a) Express the probability that at least one of these three events occurs in terms of the pi ’s. b) Express the probability that at least two of the events occur. c) Suppose that A1, A2 and A3 are independent events. Verify that P(A1|A2 ∩ A3) = P(A1). 2. Relevant equations 3. The attempt at a solution For part c I had no trouble obtaining a solution: P(A1|A2 ∩ A3) = P[A1 ∩ (A2 ∩ A3)] P(A2 ∩ A3) By independence, we have P[A1 ∩ (A2 ∩ A3)] = P(A1)P(A2)P(A3) and P(A2 ∩ A3) = P(A2)P(A3), and the result follows. However, I am having some trouble with the first two. The answer given by the instructor for a is: 1 − (1 − p1)(1 − p2)(1 − p3) and for b is: p1p2(1 − p3) + p1(1 − p2)p3 + (1 − p1)p2p3 + p1p2p3. I am not trying to dispute these answers, I am just having trouble understanding where they come from. Specifically, I do not understand the expression (1-p1), I assume that this is the probability that either p2 or p3 occur, but I'm not sure why. I was hoping somebody could give me some sort of explanation. Thanks.