Question

The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.60. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2 ) = 0.05.Are A1 and A2 mutually exclusive? Explain.

Answer 1

A1 and A2 are mutually exclusive events because the probability of A1 and A2 occurring together is 0, which meets the definition of mutually exclusive events.

Step-by-step explanation:

Based on the information provided, A1 and A2 are indeed mutually exclusive events. This is because the probability of A1 and A2 occurring together, P(A1 ∩ A2), is given as 0. When two events are mutually exclusive, this means they cannot happen at the same time, which aligns with the definition that P(A AND B) = 0 for mutually exclusive events. The prior probabilities P(A1) = 0.40 and P(A2) = 0.60 only tell us the likelihood of each event occurring individually, not about their exclusivity. Therefore, the key piece of information confirming that A1 and A2 are mutually exclusive is the fact that P(A1 ∩ A2) = 0.