Geometric Probability

Sixty-two problems in sixty-one chapters 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.

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 sixty-two classical problems in sixty-one chapters, each with an analytic solution. 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 1864, Cauchy 1841, the attribution is recorded in the chapter or bibliography.

Each chapter opens with a problem, explains the geometry, and closes with a Python simulation. Some longer classical computations are quoted or sketched with references. The code uses NumPy, with SciPy needed for two examples. Simulations agree with the analytic answers within sampling error; the printed decimal places are not a guarantee of that accuracy.