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# Buffon-Laplace Needle Problem

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

The probability is given by

 (1)

where

 (2)

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

 (3)

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 so that and , then the probabilities of a needle crossing 0, 1, and 2 lines are

 (4) (5) (6)

Defining as the number of times in tosses that a short needle crosses exactly lines, the variable has a binomial distribution with parameters and , where (Perlman and Wichura 1975). A point estimator for is given by

 (7)

which is a uniformly minimum variance unbiased estimator with variance

 (8)

(Perlman and Wishura 1975). An estimator for is then given by

 (9)

This has asymptotic variance

 (10)

which, for , becomes

 (11) (12)

(OEIS A114602).

A set of sample trials is illustrated above for needles of length , 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 , , , and a needle with length less than the shortest altitude is thrown, the probability that the needle is contained entirely within one of the triangles is given by

 (13)

where , , and are the angles opposite , , and , respectively, and is the area of the triangle. For a triangular grid consisting of equilateral triangles, this simplifies to

 (14)

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

Buffon's Needle Problem, Clean Tile Problem

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## References

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.

## Referenced on Wolfram|Alpha

Buffon-Laplace Needle Problem

## Cite this as:

Weisstein, Eric W. "Buffon-Laplace Needle Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Buffon-LaplaceNeedleProblem.html