Below I feel that I am not understanding something subtle about independence in probability relating to the influence of new information/evidence….
To give context:
Question:
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,
$$P(A|M)= \frac{P(M|A)P(A)}{P(M|A)P(A)+P(M|B)P(B)} = \frac{1/2}{1/2 + (1/10)(1/2)} = \frac{10}{11}.$$
(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 don't understand the $P(M|B)=1/10$ part exactly. When I attempted the question with writing $P(M|A^\mathsf{c})=1/10$, as $A^\mathsf{c}=B$ (if $A$ is not guilty that would that mean $B$ as they impy each other).
In writing so, how could one say that $M$ is independent of $A^\mathsf{c}$ entirely? Wouldn't knowing that person $A$ is not guilty give us information about how likely it is for person $A$ to have a matching type with the guilty party? Ie. it would be more in general to have a matching blood type given that person $A$ is innocent as $A$ was a suspect in the ordeal
(potentially important: I also thought about how person $A$ was a suspect in the ordeal, is that important too? As wouldnt being a suspect alter the chances of matching blood types entirely compared to the general population?)
Best Answer
In other words, there is some blood type that is found in $10\%$ of the population, call it blood type X. So, the chance that some particular person has blood type X is $10\%$.
Now consider $P(M|B)$. $B$ means that suspect B is in fact guilty. Because B is guilty, we know that B's blood type is X ("it is found that the guilty party had a blood type found in $10\%$ of the population"). Recall, $M$ is the event that suspect A's blood type matches the guilty party. Well, it is given that B is the guilty party and B's blood type is X. In other words,
Given that B is guilty and has blood type X, then asking for the chance that A matches B is the same as asking for the chance that A has blood type X. This is $10\%$.
Further
\begin{align*} P(M) &= P(\text{A is guilty, and A matches A})+P(\text{B is guilty, and A matches B})\\ &= \frac{1}{2}\cdot 1+\frac{1}{2}\cdot\frac{1}{10}\\ &= \frac{11}{20}\\ &\neq P(M|B) \end{align*}