Let M be the event that A's blood type matches the guilty party's. Problem says that A does match guilty party's blood type. Why isn't P(M) = 1?

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? Solution: 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)=P(M|A)P(A)P(M|A)P(A)+P(M|B)P(B)=1/21/2+(1/10)(1/2)=1011.

(We have P(M|B)=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.)

I understand that one way of deriving the P(M) is to use the Law of Total Probability, conditioning on A's guilt and B's guilt but doesn't the facts of the problem already preclude that by establishing that "Suspect A does match this blood type"?

• The probability that $A$ is guilty given that his blood type matches the guilty party is $1011$? Wow! he is not only guilty, he is over 1000 times guilty! (Note the "edit" link in the middle of the links on the lower left. Please edit your equation to make sense. It currently is gibberish.) – Paul Sinclair Mar 14 at 1:25