X positive given the sum is positive

Let $X, Y \sim \mathcal N(0,1)$ be independent. Compute $\mathbb{P}[\,X > 0 \mid X + Y > 0\,]$ .

Show hints (2)+

By rotational symmetry the direction angle is uniform. Turn both events into angle ranges.
$X+Y>0$ is $180°$ ; intersect with $X>0$ ( $135°$ ) and divide.

Answer

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0.75

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