P(A|C)=P(A|B)*P(B|C)?

Question

Is this formula P(A|C)=P(A|B)*P(B|C) correct according to Bayes' theorem? I don't think it correct but for a transition matrice system we can have something like thistransition matrice So I don't know how to prove P(A|C)=P(A|B)*P(B|C) in transition matrice though it looks very intuitive

Answer 1

The correct version of this formula would be $$ P(A \mid C) = \frac{P(A\mid B) P(B \mid C)P(C) + P(A \mid B^c)P(B^c \mid C)P(C)}{P(C)} $$ which is, more concisely, $$ P(A \mid C) = \frac{P(A \text{ and } C)}{P(C)} $$