I would like to compute the probability of $\mathbb{P}[Y > max(X_i)], Y\sim N(0, 1), X_i \sim N(0, \sigma_i)$

All the random variables have zero mean, but the variances are different.

My approaches so far were unsuccessful. I tried looking at events $A_i = P(Y>X_i)$ and their intersections and unions. But that didn't work out.

Then I took a step back and tried to search for related results online. If the $X_i$ were IID, this would be comparing Y to the n-th order statistic of a normal random variable. This question seems related: https://stats.stackexchange.com/questions/9001/approximate-order-statistics-for-normal-random-variables It seems as if even for the simplified case, there is no closed form solution for the order statistic.

Am I missing a trick here. Is it possible to calculate this in closed form?