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 parallellines
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

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).

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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. Monthly81, 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.