My textbook claims the following probability derivation, but I am still having trouble understanding where the formula came from. The derivation is as follows:
P(A|B) = P(C|B)*P(A|C) + [1-P(C|B)] * P(A|C')
Can anyone explain how this formula was derived?
It's not true in general, but correct if $A$ is independent of $B$ given $C$, i.e. $P(A|B,C)=P(A|B)$ and $P(A|B,C')=P(A|C')$. The original equation derives from total probability law:
$$P(A|B)=P(C|B)P(A|B,C)+P(C'|B)P(A|B,C')$$
If above conditions are assumed, you obtain the same equation. Note that $P(C'|B)=1-P(C|B)$.
