Let $A_1,A_2,A_3$ be events that are conditionally independent given $B$. Assume that $P(A_i\mid B)=\frac{1}{4}, P(A_i\mid B^c)=\frac{1}{2}$ for $i=1,2,3$ and $P(B)=\frac{1}{3}$.
Calculate $P(A_i\cap A_2^c\cap A_3\mid B)$ and $P(A_i\cap A_2^c\cap A_3)$.
I have calculated $P(A_i\cap A_2^c\cap A_3\mid B) = \frac{3}{64}$ since $A_1,A_2,A_3$ be events that are conditionally independent given $B$. But how to deal with $P(A_i\cap A_2^c\cap A_3)$ when $A_1,A_2,A_3$ are not conditionally independent given $B^c$ ? I try to use Bayes theorem and law of total probability but it seems I always go back to the origin and get identical equations.