For any events A,B,C is the following true? Is the following statement true? How? I'm having trouble seeing whether not it is true or false.
$$P(A\mid B) = P(A\mid B \cap C)P(C\mid B) + P(A\mid B \cap C')P(C'\mid B)$$

Here's a handy device: Write (1) $P_B(A)$ in place of $P(A\mid B)$ and (2) $P_B(A\mid C)$ in place of $P(A\mid B\cap C)$. Then the identity in question becomes clearer:
$$
P_B(A) = P_B(A\mid C) P_B(C) + P_B(A\mid C') P_B(C'),
$$
which is true because $P_B$ is a probability.
You can think of moves (1) and (2) as mentally erasing the conditioning on $B$. (Note that move (2) is justified by (1) and the definition of conditional probability.) Once you remember this device it becomes a lot easier to prove, or derive, similar expressions from other identities that you know to be true.
