This is related to my other question Renewal Theory: Probability of residual lifetime $\gamma_t > x$ conditioned on current lifetime $\delta_{t + x/2}$
$X, Y$ are continuous random variables, possibly not independent, and we know the $CDF$ of $X + Y$ which is given by $F(c)$. What is $P(X > a | Y = b)$ in terms of $F(c)$?
Attempt:
$P(X > a | Y = b) = P(X + b > a + b | Y = b) = \frac{P(X + Y > a + b)}{P(Y = b)}$. The numerator is just $F(a + b)$, but how do I find the denominator in terms of $F(c)$? Is the denominator not just $0$ since $Y$ is a continuous random variable?