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

$$P(A|B\cap C)P(C|B)=\frac{P(A\cap B\cap C)}{P(B\cap C)}\cdot{\frac{P(B\cap C)}{P(B)}}=\frac{P(A\cap B\cap C)}{P(B)} \\ P(A|B\cap C')P(C'|B)=\frac{P(A\cap B\cap C')}{P(B\cap C')}\cdot\frac{P(B\cap C')}{P(B)}=\frac{P(A\cap B\cap C')}{P(B)}$$Hence RHS is $$\frac{P(A\cap B\cap C)}{P(B)}+\frac{P(A\cap B\cap C')}{P(B)}=\frac{P(A\cap B)}{P(B)}=P(A|B)$$
