Finding conditional probability of a random variable

Question

Suppose $Y = X + N$ where $N \sim N(0,\sigma^2)$ and $X\in\{-1 , 1\}$ and $P(X = -1) = 3P(X=+1) = p$.

Calculate $F_Y(y|X=1)$ and $F_Y(y|X = -1)$.

My try: I'm confused about intersection of two events and conditional probability. Which one is correct? $$F_Y(y|X=1) = P(Y\le y|X=1) = P(N\le y-1) =F_N(y-1)$$ Or $$ F_Y(y|X=1) = P(Y\le y|X=1) = \frac{P(Y\le y\cap X=1)}{P(X=1)} = \frac{F_N(y-1)}{p/3}$$

Answer 1

Assuming independence of $X$ and $N$ we have $P(Y\leq y|X=1) =\frac {P(Y \leq y, X=1)} {P(X=1)}=\frac {P(N \leq y-1)P(X=1)} {P( X=1)}=P(N \leq y-1)$.

Without independence you cannot compute this probability.