The Buffon-Laplace needle problem asks to find the probability 
that a needle of length 
will land on at least one line, given
a rectangular grid formed by two perpendicular
families of equally spaced parallel lines,
with spacings 
and 
and needle length 
. The needle's midpoint
is uniformly distributed in a grid rectangle,
independently of its uniform
orientation. Its midpoint has coordinates 
and its orientation has 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)
|
For a positive integer 
, let 
be the number of times in 
independent tosses
that the needle crosses exactly 
lines. The random
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 Wichura 1975). For 
, the reciprocal estimator 
is
 |
(9)
|
Set 
when 
.
This has asymptotic variance
 |
(10)
|
which, for 
, becomes
 |  |  |
(11)
|
 |  |  |
(12)
|
(OEIS A114602).
A set of sample trials is illustrated above for needles with 
, 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 
, 
, 
, a needle with length 
less than the shortest altitude
is thrown with its midpoint uniformly
distributed in a tile and an independent uniform orientation. Then 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).
