What is the probability?

Question

A life insurance salesman finds that, of all the sales he makes, 70% are to people who already own policies. He also finds that, of all contacts for which no sale is made, 50% already own life insurance policies. Furthermore, 40% of all contacts result in sales. What is the probability that a sale will be made to a contact who already owns a policy?

Answer 1

Let Sale = S. Let Repeat Call = R

Then, P(R|S) = .7 means P(RS)/P(S) = .7

P(R|S') = .5 means P(RS')/P(S') = .5

P(S) = .4

Thus, P(S') = .6

P(RS) = P(S)P(R|S) = .7 * .4 = .28

P(RS') = P(S')P(R|S') =.6 * .5 = .3

P(R) = P(RS) + P(RS') = .28 + .3 = .58

Then, P(S|R) = P(RS)/P(R) = .28/.58 = 14/29 = .4828