Given a group of $n$ probability distributions, $P_1, P_2, \ldots, P_n$, we sample an outcome for each of the distributions $X_i \sim P_i, \forall i \in [n]$, and we want to compute the probability of each specific ordering of $X_i$'s.
For example, we want to compute $$ \Pr[X_1 \leq X_2 \leq \cdots \leq X_n]. $$
Also, we want to find the ordering that maximizes the probability. When the distributions are normal-distribution-like (higher probability when nearer the mean value), I can see that the probability is maximized when the distributions are sorted according to their mean values.
For more general cases, I do not have any specific ideas yet.
Any help or idea on more general cases is highly appreciated!