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Buffon's Needle Problem


BuffonNeedle

Buffon's needle problem asks to find the probability that a needle of length l will land on a line, given a floor with equally spaced parallel lines a distance d apart. The problem was first posed by the French naturalist Buffon in 1733 (Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777 (Buffon 1777, pp. 100-104).

Define the size parameter x by

 x=l/d.
(1)

For a short needle (i.e., one shorter than the distance between two lines, so that x=l/d<1), the probability P(x) that the needle falls on a line is

P(x)=int_0^(2pi)(l|costheta|)/d(dtheta)/(2pi)
(2)
=(2l)/(pid)int_0^(pi/2)costhetadtheta
(3)
=(2l)/(pid)
(4)
=(2x)/pi.
(5)

For x=l/d=1, this therefore becomes

 P(x=1)=2/pi=0.636619...
(6)

(OEIS A060294).

For a long needle (i.e., one longer than the distance between two lines so that x=l/d>1), the probability that it intersects at least one line is the slightly more complicated expression

 P(x)=2/pi(x-sqrt(x^2-1)+sec^(-1)x),
(7)

where (Uspensky 1937, pp. 252 and 258; Kunkel).

BuffonsNeedleProbability

Writing

 P(x)={(2x)/pi   for x<=1; 2/pi(x-sqrt(x^2-1)+sec^(-1)x)   for x>1
(8)

then gives the plot illustrated above. The above can be derived by noting that

 P(x)=int_0^(phi/2)int_(lsinphi/2)f_sf_phidsdphi,
(9)

where

f_s={2/d for 0<=x<=1/2d; 0 for x>1/2d
(10)
f_phi=2/pi
(11)

are the probability functions for the distance s of the needle's midpoint s from the nearest line and the angle phi formed by the needle and the lines, intersection takes place when 0<=s<=(lsinphi)/2, and phi can be restricted to [0,pi/2] by symmetry.

Let N be the number of line crossings by n tosses of a short needle with size parameter x. Then N has a binomial distribution with parameters n and 2x/pi. A point estimator for theta=1/pi is given by

 theta^^=N/(2xn),
(12)

which is both a uniformly minimum variance unbiased estimator and a maximum likelihood estimator (Perlman and Wishura 1975) with variance

 var(theta^^)=theta/(2n)(1/x-2theta),
(13)

which, in the case x=1, gives

 var(theta^^)=(theta^2(1-2theta))/(2thetan).
(14)

The estimator pi^^=1/theta^^ for pi is known as Buffon's estimator and is an asymptotically unbiased estimator given by

 pi^^=(2xn)/N,
(15)

where x=l/d, n is the number of throws, and N is the number of line crossings. It has asymptotic variance

 avar(pi^^)=(pi^2)/(2n)(pi/x-2),
(16)

which, for the case x=1, becomes

avar(pi^^)=(pi^2(1/2pi-1))/n
(17)
 approx (5.6335339)/n
(18)

(OEIS A114598; Mantel 1953; Solomon 1978, p. 7).

BuffonNeedleTosses

The above figure shows the result of 500 tosses of a needle of length parameter x=1/3, where needles crossing a line are shown in red and those missing are shown in green. 107 of the tosses cross a line, giving pi^^=3.116+/-0.073.

BuffonTosses

Several attempts have been made to experimentally determine pi by needle-tossing. pi calculated from five independent series of tosses of a (short) needle are illustrated above for one million tosses in each trial x=1/3. For a discussion of the relevant statistics and a critical analysis of one of the more accurate (and least believable) needle-tossings, see Badger (1994). Uspensky (1937, pp. 112-113) discusses experiments conducted with 2520, 3204, and 5000 trials.

The problem can be extended to a "needle" in the shape of a convex polygon with generalized diameter less than d. The probability that the boundary of the polygon will intersect one of the lines is given by

 P=p/(pid),
(19)

where p is the perimeter of the polygon (Uspensky 1937, p. 253; Solomon 1978, p. 18).

A further generalization obtained by throwing a needle on a board ruled with two sets of perpendicular lines is called the Buffon-Laplace needle problem.


See also

Buffon-Laplace Needle Problem, Clean Tile Problem

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References

Badger, L. "Lazzarini's Lucky Approximation of pi." Math. Mag. 67, 83-91, 1994.Bogomolny, A. "Buffon's Noodle." http://www.cut-the-knot.org/Curriculum/Probability/Buffon.shtml.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 139, 2003.Buffon, G. Editor's note concerning a lecture given 1733 by Mr. Le Clerc de Buffon to the Royal Academy of Sciences in Paris. Histoire de l'Acad. Roy. des Sci., pp. 43-45, 1733.Buffon, G. "Essai d'arithmétique morale." Histoire naturelle, générale er particulière, Supplément 4, 46-123, 1777.Diaconis, P. "Buffon's Needle Problem with a Long Needle." J. Appl. Prob. 13, 614-618, 1976.Dörrie, H. "Buffon's Needle Problem." §18 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 73-77, 1965.Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32, 1-37, 1995.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 209, 1998.Isaac, R. The Pleasures of Probability. New York: Springer-Verlag, 1995.Kendall, M. G. and Moran, P. A. P. Geometrical Probability. New York: Hafner, 1963.Klain, Daniel A. and Rota, G.-C. Introduction to Geometric Probability. New York: Cambridge University Press, 1997.Kraitchik, M. "The Needle Problem." §6.14 in Mathematical Recreations. New York: W. W. Norton, p. 132, 1942.Kunkel, P. "Buffon's Needle." http://whistleralley.com/buffon/buffon.htm.Mantel, L. "An Extension of the Buffon Needle Problem." Ann. Math. Stat. 24, 674-677, 1953.Morton, R. A. "The Expected Number and Angle of Intersections Between Random Curves in a Plane." J. Appl. Prob. 3, 559-562, 1966.Perlman, M. and Wichura, M. "Sharpening Buffon's Needle." Amer. Stat. 20, 157-163, 1975.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Schuster, E. F. "Buffon's Needle Experiment." Amer. Math. Monthly 81, 26-29, 1974.Sloane, N. J. A. Sequences A060294 and A114598 in "The On-Line Encyclopedia of Integer Sequences."Solomon, H. "Buffon Needle Problem, Extensions, and Estimation of pi." Ch. 1 in Geometric Probability. Philadelphia, PA: SIAM, pp. 1-24, 1978.Stoka, M. "Problems of Buffon Type for Convex Test Bodies." Conf. Semin. Mat. Univ. Bari, No. 268, 1-17, 1998.Uspensky, J. V. "Buffon's Needle Problem," "Extension of Buffon's Problem," and "Second Solution of Buffon's Problem." §12.14-12.16 in Introduction to Mathematical Probability. New York: McGraw-Hill, pp. 112-115, 251-255, and 258, 1937.Wegert, E. and Trefethen, L. N. "From the Buffon Needle Problem to the Kreiss Matrix Theorem." Amer. Math. Monthly 101, 132-139, 1994.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 53, 1986.Wood, G. R. and Robertson, J. M. "Buffon Got It Straight." Stat. Prob. Lett. 37, 415-421, 1998.

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Buffon's Needle Problem

Cite this as:

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

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