I am solving the Strategic Practice and Homework Ex#2 of the course Stat 110 by Harvard University (I am not a student there, I am practicing Probability). I have a conceptual doubt in one of the questions. Here's the question as it is:
(Q4, Practice 2) 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.
In the solution to this question we have:
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”
As per the solution we have $P(\frac{M}{A})=1$ and my doubt is that as we already know that the blood group of A matches with the particular blood type discussed in the question, so shouldn't $P(M)=1$ because we already know that A has this particular blood group. I am not getting error with this reasoning and it has to be wrong because $P(M)≠P(\frac{M}{A})$.
Edit:
It's been pointed out in one of the comments that here $P(M) = P(\frac{M}{A})$ but I am not quite sure if that is right because, part of the solution of this question is the following image: Part of the solution to calculate P(A|M) Here we can see that the $P(\frac{A}{M}) = \frac{10}{11}$ but if we assume that $P(\frac{M}{A}) = P(M)$ then it would be $\frac{1}{2}$ because $P(\frac{A}{M}) = P(\frac{M}{A})*\frac{P(A)}{P(M)} = P(A) = \frac{1}{2}$ which doesn't look right.
