Bibliography

Foundational monographs

N. Alon & J. H. Spencer, The Probabilistic Method, 4th ed., Wiley, 2016. Advanced applications of probabilistic reasoning in combinatorics.

H. A. David & H. N. Nagaraja, Order Statistics, 3rd ed., Wiley, 2003. The definitive reference for the order-statistic identities underlying Chapters 48 and 38.

M. Kendall & P. A. P. Moran, Geometrical Probability, Griffin, 1963. The classical mid-twentieth-century text on the subject; the source for many of the constants discussed here.

D. A. Klain & G.-C. Rota, Introduction to Geometric Probability, Cambridge University Press, 1997. A modern and unusually accessible treatment emphasising integral geometry and invariant measures.

L. A. Santaló, Integral Geometry and Geometric Probability, Cambridge University Press, 1976. The magnum opus on the integral-geometric side.

H. Solomon, Geometric Probability, Society for Industrial and Applied Mathematics, 1978. An essential supplement to Kendall–Moran; particularly strong on random-convex-set and line-arrangement results.

A. M. Mathai, An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon & Breach, 1999.

Primary sources cited

J. Bertrand, Calcul des probabilités, Gauthier-Villars, 1889. The origin of Bertrand’s paradox of the random chord.

W. Blaschke, “Über affine Geometrie XI,” Leipzig Ber. Math.-Phys. Kl. 69 (1917), 412–420. Sylvester’s four-point problem: extremal shapes.

G.-L. L. de Buffon, Essai d’arithmétique morale, supplement to Histoire Naturelle, 1777. The needle problem; also contains the franc-carreau coin tile problem of 1733.

A.-L. Cauchy, “Mémoire sur la rectification des courbes et la quadrature des surfaces courbes,” Mém. Acad. Sci. Paris 22 (1841), 3–15. The shadow/projection formula.

E. Czuber, Geometrische Wahrscheinlichkeiten und Mittelwerte, Teubner, 1884. First computation of the disc line-picking constant $128/(45\pi)$.

F. Mosteller, Fifty Challenging Problems in Probability, Dover, 1965 (reissued 1987). A classic collection in the same spirit as this volume.

D. P. Robbins, “Average distance between two points in a box,” American Mathematical Monthly 85 (1978), 278. The unit-cube constant.

W. L. Stevens, “Solution to a geometrical problem in probability,” Annals of Eugenics 9 (1939), 315–320. The covering formula for random arcs.

J. J. Sylvester, Educational Times (April 1864), question 1491. The four-point problem.

P. Valtr, “Probability that $n$ random points are in convex position,” Discrete & Computational Geometry 13 (1995), 637–643. Convex-position formulas in planar regions.

J. G. Wendel, “A problem in geometric probability,” Mathematica Scandinavica 11 (1962), 109–111. The sign-flip argument for $n$ points on a sphere.

W. S. B. Woolhouse, “Problem 2059,” Educational Times (1867). Expected triangle area in a disc.