This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
Given two random points x and y on the interval 0..1, what is the average
size of the smallest of the three resulting intervals?
In between these positions the surface forms a series of planes.
Thus the volume under it consists of 2 pyramids each with an
altitude of 1/3 and an (isosceles triangular) base of area 1/2,
yielding a total volume of 1/9.