Geometric Probability
Chance and the plane
2026 · 153 pp
Forty-eight problems in classical geometric probability, the broken stick, Buffon's needle, Bertrand's paradox, Sylvester's four-point problem, Wendel's theorem, Cauchy's shadow formula, each with an analytic solution and a Monte Carlo verification box.
Geometric probability is the branch of the subject where the sample space is a region of the plane, a disc, or a higher-dimensional ball, rather than a finite set of outcomes. Three sticks, two chords, Buffon’s needle on a grid of lines, a random triangle inside a square, the convex hull of random points on the sphere: the answers are almost always a surprise.
This volume collects forty-eight classical problems, each with an analytic solution and a Monte Carlo verification box in Python. Topics range from the broken-stick triangle and Bertrand’s paradox through Wendel’s theorem on the plane and the sphere, Sylvester’s four-point problem, Cauchy’s formula for the expected shadow of a convex body, and Buffon’s noodle. Where a result is attributable, Wendel 1962, Sylvester 1865, Cauchy 1832, the attribution is recorded at the head of the chapter.
Each chapter opens with the problem, sets up the coordinates, derives the answer, and closes with a short Python program that reproduces the value to four decimal places.