Fall 2011 Strategic Practice 2: Section 3 (Thinking Conditionally) - Question 4
A crime is committed by one of two suspects, A and B. Initially, there is equal evidence against both of them. In further investigation at the crime scene, it is found that the guilty party had a blood type found in 10% of the population. Suspect A does match this blood type, whereas the blood type of Suspect B is unknown.
(a) Given this new information, what is the probability that A is the guilty party?
(b) Given this new information, what is the probability that B's blood type matches that found at the crime scene?
Solution: (a) Let M be the event that A's blood type matches the guilty party's and for brevity, write A for A is guilty‚ and B for B is guilty. By Bayes' Rule, P(A|M) = 10/11. Note that we have P(M|B) = P(M) = 1/10 since, given that B is guilty, the probability that A's blood type matches the guilty party's is the same probability as for the general population. (b) Let C be the event that B's blood type matches, and condition on whether B is guilty. This gives P(C|M) = P(C|M, A) P(A|M) + P(C|M, B) P(B|M) = 2/11.