For two independent Uniform(0,1) draws, what are E[max] and E[min], and what is the general result for n draws?

How to answer

E[max] = 2/3 and E[min] = 1/3 for two draws. By symmetry E[min] + E[max] = E[X] + E[Y] = 1, and each is equidistant from the center, so E[min] = 1/3, E[max] = 2/3. Generally, for n iid U(0,1), E[max] = n/(n+1) and E[min] = 1/(n+1) — the order statistics divide [0,1] into n+1 equal expected gaps.

Key idea: n+1 expected equal gaps; max sits at n/(n+1).