A sample is selected from one of two populations, S1 and S2, with probabilities P(S1) = .7 and P(S2) = .3. If the sample has been selected from S1, the probability of observing an event A is P(A | S1) = .2. Similarly, if the sample has been selected from S2, the probability of observing A is P(A | S2) = .3.
a. If a sample is randomly selected from one of the two populations, what is the probability that event A occurs?
b. If the sample is randomly selected and event A is observed, what is the probability that the sample was selected from population S1? From population S2?