A half-plane, given positive

Let $X, Y$ be independent standard normal variables. Compute $\mathbb{P}\big[\,Y > \sqrt{3}\,X \ \big|\ Y > 0\,\big]$ .

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The joint density is rotationally symmetric, so the direction angle is uniform - probabilities are angle fractions.
$Y>0$ is $180°$ ; $Y>\sqrt3 X$ is $\Theta\in(60°,240°)$ . Intersect and divide.

Answer

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0.6667 (± 0.001)

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