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, with 
. The needle's midpoint
is uniformly distributed modulo the line
spacing, independently of its uniform
orientation. The problem was first posed by the French naturalist Buffon (Buffon
1733, pp. 43-45), and reproduced with solution by 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 
is the inverse secant on 
(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 independent probability density 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]](/images/equations/BuffonsNeedleProblem/Inline33.svg)
by symmetry.
Let 
be the number of line crossings in 
independent tosses
of a needle with 
, where 
is a positive integer.
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 Wichura 1975) with variance
 |
(13)
|
which, in the case 
, gives
 |
(14)
|
For 
, the reciprocal estimator 
for 
, known as Buffon's estimator, is given by
 |
(15)
|
where 
, 
is the number of throws, and 
is the number of line crossings.
Set 
when 
. With this convention, it is asymptotically unbiased,
meaning that its expectation value tends to
, and converges
in probability to 
as 
. 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 
. Substituting
this estimate into the asymptotic variance formula gives
an approximate standard error of 0.267.
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 with 
. 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 
, using independent uniform translation modulo the line spacing
and uniform orientation. 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.
." Math. Mag. 67,
83-91, 1994.Bogomolny, A. "Buffon's Noodle." 
." Ch. 1 in