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.
$$P(A\mid B) = P(A\mid B \cap C)P(C\mid B) + P(A\mid B \cap C')P(C'\mid B)$$

It is true, generally speaking:
\begin{align}
P(A \mid B) & = \frac{P(A, B)}{P(B)} \\
            & = \frac{P(A, B, C)+P(A, B, C')}{P(B)} \\
            & = \frac{P(A, B, C)}{P(B)}+\frac{P(A, B, C')}{P(B)} \\
            & = \frac{P(A, B, C)}{P(B, C)} \cdot \frac{P(B, C)}{P(B)}
              + \frac{P(A, B, C')}{P(B, C')} \cdot \frac{P(B, C')}{P(B)} \\
            & = P(A \mid B, C) \cdot P(C \mid B)
              + P(A \mid B, C') \cdot P(C' \mid B)
\end{align}
Note any potential gotchas in the denominators.
