Conditional probability puzzle 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?
So here is my approach. Let A stands for "A gulity", M for "A matching the blood type" and N for "B matching the blood type". Then,
P(A) = P(B) = $\frac{1}{2}$
P(M) = $1$ 
P(N) = $\frac{1}{10}$
$P(A|M)=\frac{P(M|A)P(A)}{P(M)}$
$P(A|M)=\frac{1.\frac{1}{2}}{1}$
$=\frac{1}{2}$
The answer is given as $\frac{10}{11}$. More than arriving at this answer I am keen to understand what am I missing here.

I think the problem comes with your $M$ and $N$ events. Suppose we have the next events:
Event $A$ for A is guilty.
Event $B$ for B is guilty.
Event $F$ for blood type is found in the crime scene.
As you have stated, we have first:
$$ P(A) = P(B) = 1/2$$
We also have:
$$ P(F \vert A) = 1$$
$$ P(F \vert B) = 1/10$$
Which leads to:
$$P(F) = P(F \vert A) P(A) + P(F \vert B) P(B) = 11/20$$
And by Bayes:
$$ P(A \vert F) = \frac{P(F \vert A) P(A)}{P(F)} = 10/11$$
