Topical Guide

This book is organised into seven thematic Parts. For quick navigation, the following topical guide groups chapters by mathematical technique and by the kind of object the problem concerns.

By technique

Area ratios in parameter space. Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

Linearity of expectation. Chapters 18, 22, 50, 47.

Inclusion–exclusion. Chapters 54, 9, 48, 19.

Symmetry and sign-flips (Wendel’s theorem). Chapters 25, 26, 27, 29, 30, 14.

Polar-coordinate integration. Chapters 31, 35, 34, 37, 39, 41.

Order statistics. Chapters 17, 48, 38, 37.

Change of variables & distributions. Chapters 13, 15, 20, 55, 28, 61.

Integral geometry / Crofton formula. Chapters 21, 58.

Number theory (Euler product). Chapters 52, 56.

By object

Broken stick. Chapters 1, 2, 7, 48, 9.

Random chords. Chapters 12, 13, 15, 16, 18, 20, 21, 22, 23, 24.

Random triangles. Chapters 25, 26, 43, 44, 45, 46, 47, 49, 30, 50, 51.

Random points in the disc. Chapters 31, 35, 37, 39, 41, 49, 47, 30.

Random points in a square. Chapters 3, 32, 44, 45, 34, 10, 36, 40.

Random points on a circle. Chapters 12, 25, 26, 13, 14, 15, 17, 18, 22, 50, 51, 24.

Random points on a sphere. Chapters 27, 28, 29, 61.

Random points in cubes and balls. Chapters 59, 60.

Buffon-family problems. Chapters 53, 6, 54, 57.

Classical constants. Values $\tfrac14$: Chapters 25, 26, 30. $\tfrac23$: Chapters 4, 43, 15. $\tfrac{1}{n^2}$: Ch. 17. $\tfrac{4}{\pi}$: Ch. 13. $\tfrac{6}{\pi^2}$: Ch. 52. $\tfrac{128}{45\pi}$: Ch. 35. $\tfrac{35}{48\pi}$: Ch. 49. $\tfrac{36}{35}$: Ch. 59. Robbins’s constant: Ch. 60.