I can do n=1 (the probability is 1, obviously) and n=2 (the probability is 12+1πsin−1r, not so obviously). n=3 and up seem harder, and my pattern-spotting skills are not sufficient to intuit the general case from those two :-).
Heh. I’ve sometimes thought it’d be nice to have a copy of Eureqa or the other symbolic tools, to feed the Monte Carlo results into and see if I could deduce any exact formula given their hints. I don’t need exact formulas often but it’s nice to have them. I’ve noticed people can do apparently magical things with Mathematica in this vein. All proprietary AFAIK, though.
I can do n=1 (the probability is 1, obviously) and n=2 (the probability is 12+1πsin−1r, not so obviously). n=3 and up seem harder, and my pattern-spotting skills are not sufficient to intuit the general case from those two :-).
Heh. I’ve sometimes thought it’d be nice to have a copy of Eureqa or the other symbolic tools, to feed the Monte Carlo results into and see if I could deduce any exact formula given their hints. I don’t need exact formulas often but it’s nice to have them. I’ve noticed people can do apparently magical things with Mathematica in this vein. All proprietary AFAIK, though.
Writeup: https://www.gwern.net/Order-statistics#probability-of-bivariate-maximum