Solution
The Average Chord
Show the problem again
Two points are chosen independently and uniformly at random on a circle of radius 1. What is the expected straight-line distance between them? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is 4/π ≈ 1.2732. By rotational symmetry fix the first point; the central angle θ to the second point is uniform on [0, 2π]. The chord subtending angle θ has length 2 sin(θ/2). So E[distance] = (1/2π) ∫₀^{2π} 2 sin(θ/2) dθ = (1/2π) · [−4 cos(θ/2)]₀^{2π} = (1/2π) · 8 = 4/π. That is about 64% of the diameter, noticeably more than half: random chords are biased toward being long, because the sine is flat near its peak.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.