Conditional probability problem with Bayes' Rule The question is as follows: > 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. Given this new information, what is the probability that A is the guilty party? My solution: > Let $A$ be the event that A is guilty. Let $B$ be the event that B is guilty. Let $C$ be the event of having the guilty blood type. It follow that: $$P(A|C)=\frac{P(C|A)P(A)}{P(C)}=\frac{(1)(0.5)}{(0.1)}=5$$ Does anyone know what is wrong with this approach?

The last probability should be

$$P(A|C)=\frac{P(C|A)P(A)}{P(C)}=\frac{P(C|A)P(A)}{P(C|A)P(A) + P(C|B)P(B)}$$

where $P(A) = P(B) = \frac{1}{2}$ as one of the suspects $A$ or $B$ did it, a priori.

We know $P(C|A) = 1$, as we measured the blood type of $A$.

We don't know $B$'s so we must assume $P(C|B) = \frac{1}{10}$ as this is the population fraction.

Now filling these in I get $\frac{10}{11}$.