Solution
Cutting Off a Segment
Show the problem again
Two points are chosen independently and uniformly on the boundary of a disk, and the chord joining them cuts the disk into two pieces. What is the expected area of the smaller piece, as a fraction of the disk's area? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is 1/4 − 1/π² ≈ 0.1487. Take the radius to be 1. Fix the first point; the second is at a uniform angle φ ∈ [0, 2π), and the minor arc has central angle θ = min(φ, 2π − φ), which is uniform on [0, π]. The smaller piece is a circular segment of area (θ − sin θ)/2. So E[area] = (1/π) ∫₀^π (θ − sin θ)/2 dθ = (1/2π)(π²/2 − 2) = π/4 − 1/π. Dividing by the disk's area π gives 1/4 − 1/π². Note that this is well below 1/4: the smaller piece tends to be thin, since the segment area grows like θ³ for small angles.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.