This is a fairly basic question, but I'm not exactly sure what the answer to it is:
If $P(A)=0.9$ , $P(B \mid A)=0.8$ and $P(B \mid A^C)=0.7$, what is the probability of $P(A \mid B)$ ?
I can't plug it into the standard formula for conditional probability since $P(B)$ is not known. My initial thought was that the probability of A given B is equal to the probability of A ($0.9$) because the probability of A is seemingly unaffected by the occurrence of B. Is my intuition correct?
$$P(B)=P(B\cap A)+P(B\cap A^{C})$$ $$=P(B|A)P(A)+P(B|A^{C})P(A^{C})$$ $$=P(B|A)P(A)+P(B|A^{C})[1-P(A)].$$