Average Distance Between Random Points on a Line Segment Suppose I have a line segment of length $L$.  I now select two points at random along the segment.  What is the expected value of the distance between the two points, and why?

Without loss of generality, let the interval length be L and the sought average difference D.
The answer is seen to be D=L/3 by the following simple argument:
Select 3 random values a,b,c on the interval of length L. The probability that c is between a and b equals 1/3 since the other equiprobable alternatives are that a or b are in the middle.  Since the interval length is L, the probability must also be D/L, thus D=L/3.
