Probability that one normal (uncorrelated) variable is greater than another if the latter is positive

Question

Assume that $X\sim N(0,\sigma_x^2)$, $Y\sim N(0,\sigma_y^2)$ and $X$ and $Y$ are uncorrelated. Can we solve analytically for $\mathbb P(X>Y |Y>0)$?

Answer 1

Hint: $\mathbb P(X>Y |Y>0) = \frac{ \mathbb P(X>Y,Y>0)}{ \mathbb P(Y>0)} = 2 \mathbb P(X>Y,Y>0)$

If $X$ and $Y$ are independent, then you can solve this easily. If not, I think you are not given enough information to solve it.