Solution
Two Points on a Globe
Show the problem again
Two points are chosen independently and uniformly at random on the surface of a sphere of radius 1. What is the expected straight-line (chord) distance between them? Give a reduced fraction.
Worked solution
The answer is 4/3. By symmetry place the first point at the north pole (0, 0, 1). For the second point (x, y, z), the squared distance is x² + y² + (1 − z)² = 2 − 2z. By Archimedes' hat-box theorem z is uniform on [−1, 1]. Therefore E[distance] = (1/2) ∫_{−1}^{1} √(2 − 2z) dz = (1/4) ∫₀⁴ √w dw = (1/4)(2/3)(8) = 4/3, with w = 2 − 2z. Compare the circle, where the answer is 4/π ≈ 1.273: on a sphere most of the area is near the "equator" relative to the first point, at distance about √2.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.