正态分布

由于 $X$ 和 $Y$ 都是均值 $0$ 正态随机变量，因此 $Y-4X$ 也是均值 $0$ 正态随机变量。因此，$P[Y > 4X] = P[Y-4X > 0]$ 询问的是正态随机变量大于其平均值的概率。这是 $0.5$ 的正态分布关于其均值的对称性。

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Original Explanation

Since $X$ and $Y$ are both mean $0$ normal random variables, $Y-4X$ is also a mean $0$ normal random variable. Therefore, $P[Y > 4X] = P[Y-4X > 0]$ is asking about the probability that a normal random variable is greater than its mean. This is $0.5$ by the symmetry of the normal distribution about its mean.