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

(2) (3) (4) (5)

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

(8)

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

(9)

where

(10) (11)

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 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

(17) (18)

(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.