We plot the 2-space in which the two points can be, as shown to the right. We are constrained by 0<a, a<b, b<1, giving the triangle shown in blue.
The value of the distance we are interested in is shown by the green lines. We want its mean, over the blue triangle. This will be the value of the distance at the centroid of the triangle, as marked by a black dot. The centroid of a triangle lies ⅓ of the way along any of its medians, so the answer is ⅓.
Solution using symmetry: show
Consider instead the problem
Points p, q, r are chosen, independently and at random, on a
ring of unit circumference. The ring is cut at p, straightened into a stick,
and then cut at q and r. What is the expected length of piece qr?
We have cut a ring at three independently chosen random points, to form three
pieces. The total length is 1, so the expected length of each piece must be
⅓. This problem is equivalent to the one
originally given.
Consider instead the problem
Points p, q, r are chosen, independently and at random, on a ring of unit circumference. The ring is cut at p, straightened into a stick, and then cut at q and r. What is the expected length of piece qr?
We have cut a ring at three independently chosen random points, to form three pieces. The total length is 1, so the expected length of each piece must be ⅓. This problem is equivalent to the one originally given.