My question is similar to a question answered before many times, namely the expected distance of L/3 between two random points on a line segment (Average Distance Between Random Points on a Line Segment)
However, I would like to add the prerequisite of starting by having an infinite line with infinite points on it. After picking a line segment at random and observing it contains two points, what is the expected distance between these points now?
I think the answer is L/2, since you choose the line segment after defining the points, which makes the expected length of the head and tail of the segment both L/4 and the distance between the points L/2.
My friend claims it is still L/3. Please help!