Question

The prior probabilities for events A1 and A2 are P(A1) = 0.20 and P(A2) = 0.50. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. (a) Are A1 and A2 mutually exclusive?

Answer 1

The inclusion-exclusion principle says

P(A1 ∩ A2) = P(A1) + P(A2) - P(A1 ∪ A2)

We know the probability of intersection is 0, so

P(A1) + P(A2) = P(A1 ∪ A2)

which means A1 and A2 are indeed mutually exclusive.