# Probability that maximal elements has the same position in samples from correlated random variables

Let $$x$$ and $$y$$ be two correlated random variable (say, standard normal) with correlation coefficient $$\rho>0$$. Let $$X= \{x_1, x_2, ..., x_L\}$$ and $$Y= \{y_1, y_2, .. y_L\}$$ be samples of size $$L$$ from $$x$$ and $$y$$ respectively.

What is the probability that $$\mbox{argmax}\ X = \mbox{argmax}\ Y$$.

Alternatively, suppose that $$x_1$$ is maximal element, what is the probability that $$y_1$$ is maximal too.

Any references pointing to the solution of either question would also be appreciated.