Solution
Two Marks on a Ruler
Show the problem again
Two points are chosen independently and uniformly at random on a segment of length 1. What is the expected distance between them? Give your answer as a reduced fraction.
Worked solution
The answer is 1/3. The two points cut the segment into three pieces: left, middle, right. The joint distribution of these three lengths is symmetric under any permutation of the pieces (uniform spacings are exchangeable), so each piece has the same expected length. The three lengths always sum to 1, so each has expectation 1/3, and the distance between the points is exactly the middle piece. As a check by direct integration: E|X − Y| = ∫∫ |x − y| dx dy = 2∫₀¹∫₀ˣ (x − y) dy dx = 2∫₀¹ x²/2 dx = 1/3.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.