I am trying to somewhat generalize a question, which has been asked in one way or another a several times here on StackExchange**. However, I have not managed to find an answer to the below problem.
Suppose, $X_1,\ldots,X_N\sim\mathcal{N}(\mu_X, \sigma_X)$ and $Y_1,\ldots,Y_M\sim\mathcal{N}(\mu_Y, \sigma_Y)$, which are independently drawn. We are interested in the following probability:
$$ P(\text{there exists at least one }X_i\text{ such that } X_i \text{ greater than max}\{Y_1,\ldots, Y_M\})=P(\exists X\in\{X_1,\ldots, X_N\}:X>\text{max}\{Y_1,\ldots, Y_M\})\tag{1} $$
I have tried to decompose this problem into smaller problems by writing the "there exists" part as
$$ P(\exists X\in\{X_1,\ldots, X_N\}:X>\text{max}\{Y_1,\ldots, Y_M\})=P(X_1>\text{max}\{Y_1,\ldots, Y_M\}\cup\ldots\cup X_N>\text{max}\{Y_1,\ldots, Y_M\}), $$
but the RHS does not contain any information about the partial order within the set of $\{X_i\}$. Trying to add this information to the RHS makes it overly complicated and I am not even sure if this would work out. Is there a known closed form expression or a numerical way to calculate the value in Eq.(1)?
** $P(X_1 < \min(X_i,\ldots, X_n))$ across different normal random variables