Consider a series of points distributed uniformly on the unit square, $X_i, Y_i \sim \mathcal{U}(0,1)$. I then take the subset of those points for which $Y_i \geq X_i$. Let's say there are $n$ such points afterwards.
I want to find the probability that there is at least one point $(X_k, Y_k)$ such that $\forall i \neq k$, $X_k \leq X_i$ and $Y_k \geq Y_i$. Another way to see this is that if I construct the triangle $(X_k, X_k), (X_k, Y_k), (Y_k, Y_k)$, then this triangle contains all the points. The question is how likely am I to find that such a point $(X_k, Y_k)$ exists given $n$ points in the region?
$\Pr(\exists k : k = \arg\min X_i = \arg\max Y_i)$ for $n$ points drawn uniformaly in the upper left triangle of the unit square.
I'd prefer a solution to this particular phrasing of the problem rather than a solution to an equivalent problem, but such solutions would still be nice to see.