What is the conditional probability $P(X \gt 0 \mid X + Y \gt 0)$?

Question

Suppose $X$ and $Y$ are both independent standard normal random variables,

with $X \sim {\rm N} (0; 1)$ and $Y \sim {\rm N} (0; 1)$.

Answer 1

We want $$\frac{\Pr((X\gt 0) \cap (X+Y\gt 0))}{\Pr(X+Y\gt 0)}.$$ By symmetry the denominator is $\frac{1}{2}$.

For the numerator, we want the probability that $(X,Y)$ lands in the part of the plane that is to the right of the $y$-axis, and above the line $y=-x$. By symmetry this is $\frac{3}{8}$.