Buffon-Laplace Needle Problem


The Buffon-Laplace needle problem asks to find the probability P(l,a,b) that a needle of length l will land on at least one line, given a floor with a grid of equally spaced parallel lines distances a and b apart, with l<a,b. The position of the needle can be specified with points (x,y) and its orientation with coordinate phi. By symmetry, we can consider a single rectangle of the grid, so 0<x<a and 0<y<b. In addition, since opposite orientations are equivalent, we can take -pi/2<phi<pi/2.

The probability is given by




(Uspensky 1937, p. 256; Solomon 1978, p. 4), giving


This problem was first solved by Buffon (1777, pp. 100-104), but his derivation contained an error. A correct solution was given by Laplace (1812, pp. 359-362; Laplace 1820, pp. 365-369).


If a=b so that x=l/a=l/b and 0<x<1, then the probabilities of a needle crossing 0, 1, and 2 lines are


Defining N_i as the number of times in n tosses that a short needle crosses exactly n lines, the variable N_1+N_2 has a binomial distribution with parameters n and m/pi, where m=x(4-x) (Perlman and Wichura 1975). A point estimator for theta=1/pi is given by


which is a uniformly minimum variance unbiased estimator with variance


(Perlman and Wishura 1975). An estimator pi^^=1/theta^^ for pi is then given by


This has asymptotic variance


which, for x=1, becomes

 approx (0.465821)/n

(OEIS A114602).


A set of sample trials is illustrated above for needles of length a/l=b/l=0.3, where needles intersecting 0 lines are shown in green, those intersecting a single line are shown in yellow, and those intersecting two lines are shown in red.

If the plane is instead tiled with congruent triangles with sides a, b, c, and a needle with length l less than the shortest altitude is thrown, the probability that the needle is contained entirely within one of the triangles is given by


where A, B, and C are the angles opposite a, b, and c, respectively, and K is the area of the triangle. For a triangular grid consisting of equilateral triangles, this simplifies to


(Markoff 1912, pp. 169-173; Uspensky 1937, p. 258).

See also

Buffon's Needle Problem, Clean Tile Problem

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Buffon, G. "Essai d'arithmétique morale." Histoire naturelle, générale er particulière, Supplément 4, 46-123, 1777.Laplace, P. S. Théorie analytique des probabilités. Paris: Veuve Courcier, 1812.Laplace, P. S. Théorie analytique des probabilités, 3rd rev. ed. Paris: Veuve Courcier, 1820.Markoff, A. A. Wahrscheinlichkeitsrechnung. Leipzig, Germany: Teubner, 1912.Perlman, M. and Wichura, M. "Sharpening Buffon's Needle." Amer. Stat. 20, 157-163, 1975.Schuster, E. F. "Buffon's Needle Experiment." Amer. Math. Monthly 81, 26-29, 1974.Sloane, N. J. A. Sequence A114602 in "The On-Line Encyclopedia of Integer Sequences."Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, pp. 3-6, 1978.Uspensky, J. V. "Laplace's Problem." §12.17 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 255-257, 1937.

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Buffon-Laplace Needle Problem

Cite this as:

Weisstein, Eric W. "Buffon-Laplace Needle Problem." From MathWorld--A Wolfram Web Resource.

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