Buffon's Needle Problem
Buffon's needle problem asks to find the probability that a needle of length 
will land on a line, given a floor with equally spaced parallel lines a distance 
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 
by
 |
(1)
|
For a short needle (i.e., one shorter than the distance between two lines, so that 
), the probability 
that the needle falls on a line is
For 
, this therefore becomes
 |
(6)
|
(OEIS A060294).
For a long needle (i.e., one longer than the distance between two lines so that 
), the probability that it intersects at least one line is the slightly more complicated expression
 |
(7)
|
where (Uspensky 1937, pp. 252 and 258; Kunkel).
Writing
![P(x)=<span style=]() {(2x)/pi for x<=1; 2/pi(x-sqrt(x^2-1)+sec^(-1)x) for x>1 " src="https://mathworld.wolfram.com/images/equations/BuffonsNeedleProblem/NumberedEquation4.gif" style="height:82px; width:277px" /> |
(8)
|
then gives the plot illustrated above. The above can be derived by noting that
 |
(9)
|
where
are the probability functions for the distance 
of the needle's midpoint 
from the nearest line and the angle 
formed by the needle and the lines, intersection takes place when 
, and 
can be restricted to ![[0,pi/2]](https://mathworld.wolfram.com/images/equations/BuffonsNeedleProblem/Inline31.gif)
by symmetry.
Let 
be the number of line crossings by 
tosses of a short needle with size parameter 
. Then 
has a binomial distribution with parameters 
and 
. A point estimator for 
is given by
 |
(12)
|
which is both a uniformly minimum variance unbiased estimator and a maximum likelihood estimator (Perlman and Wishura 1975) with variance
 |
(13)
|
which, in the case 
, gives
 |
(14)
|
The estimator 
for 
is known as Buffon's estimator and is an asymptotically unbiased estimator given by
 |
(15)
|
where 
, 
is the number of throws, and 
is the number of line crossings. It has asymptotic variance
 |
(16)
|
which, for the case 
, becomes
(OEIS A114598; Mantel 1953; Solomon 1978, p. 7).
The above figure shows the result of 500 tosses of a needle of length parameter 
, where needles crossing a line are shown in red and those missing are shown in green. 107 of the tosses cross a line, giving 
.
Several attempts have been made to experimentally determine 
by needle-tossing. 
calculated from five independent series of tosses of a (short) needle are illustrated above for one million tosses in each trial 
. 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 
. The probability that the boundary of the polygon will intersect one of the lines is given by
 |
(19)
|
where 
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.
REFERENCES:
Badger, L. "Lazzarini's Lucky Approximation of 
." 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 
." 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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