Buffon's Needle Problem
Numbers
in parentheses correspond to the numbered references in my publication
list.
INTRODUCTION
Many years ago (many!) when
I took my master's degree exam at Virginia Tech I found, to my horror,
the following problem:

The Comte de Buffon
(1707  1788)
Click on the picture for a brief biography of Buffon
and links to related information about mathematicians and mathematics.

I had no idea how to solve
this problem. Fortunately, I passed the exam anyway. However, I became
curious about the problem and set about investigating it. That investigation
was the beginning of a fascinating adventure into the area of geometric
probability  methods that combine geometry and probability. I discovered
that Georges Louis Leclerc, Comte de Buffon, lived from 1707 to 1788 in
France. The needle problem and its solution were discovered in a note in
"Actes de l'Academie des Sciences" in Paris, 1733, and Buffon published
them eventually in "Essai d'arithmetique morale" in 1777. 
The solution to the needle
problem goes as follows.
A figure can be found in
(55) (this article is
in Swedish).
GENERALIZATION
The solution to the needle problem
struck me as simple and elegant (as is typical, the solution of a problem
is simple after you know what it is!!). However, I wondered what would
happen in the case where the length of the needle is larger than the distance
between parallel lines; i.e., k > d. Of course, the probability increases.
The solution to the problem with general k, which can be found in Khamis
(1988), is given here:
Now I felt that I knew all
that was necessary to know about Buffon's needle problem. What more can
one possibly say about it?
CONCENTRIC CIRCLES
In 1983 I was invited to give
a talk at Washington and Lee University in Lexington, Virginia to the math
majors. I decided to talk about Buffon's needle problem. At the end of
the talk, one student asked about the probability of a cross if there were
concentric circles instead of parallel lines on the plane region. I responded
that I didn't know what the probability would be in that case, but that
I was sure that someone had solved the problem. When I returned to Dayton,
I searched the literature for "Buffon's needle problem on concentric circles."
I was surprised to find nothing. So, I began working on the problem. I
soon discovered why no one had solved the problem  such involved calculations!
After some persistence however:
If a needle of length k is
randomly dropped onto a board with N concentric circles, where the difference
in radii between any two consecutive such circles is a constant d, k <
d, then the probability that the needle crosses a circumference is (10):
BAYESIAN BUFFON NEEDLE PROBLEM
In 1985 I gave a talk at the
RoseHulman Conference in Terre Haute, Indiana about Buffon's needle problem
on concentric circles. At the end of the talk, a member of the audience
pointed out that it would be more practical to model the probability so
that the midpoint of the needle has a tendency to fall near the center
of the concentric circles rather than randomly throughout the plane region
containing N circles. I thought about this application when I returned
to Dayton. If we assume an a priori bivariate normal distribution
for the coordinates of the midpoint of the needle, then we can derive the
a posteriori distribution of the distance between the midpoint of
the needle and the nearest circumference. Then, with some calculation,
the P(cross) can be obtained, but cannot be written in closed form; i.e.,
it contains several integrals that can not be expressed in closed form.
See (21) for the gory
details.
VARIATIONS AND APPLICATIONS
There are many other variations
of Buffon's needle problem. What happens, for example, if a needle is randomly
dropped onto a plane surface containing n radial lines (24),
or a grid of horizontal and vertical lines (Laplace's problem)? These problems
are interesting to think about, but the methods used to solve them are
also extremely important in practical applications. The methods of geometric
probability can be used to solve problems involving cancer cells, military
strategies, virus particles, chromosome positions, etc. See Solomon
(1978) for examples.
CONCLUSION
In calendar years 19992000
I was a visiting professor at Uppsala University in Sweden. One of the
most famous professors from Uppsala University is Carolus Linneaus (1707
 1778), the father of taxonomy. His gardens are still cared for and open
to the public every summer. Buffon, who was also a noted naturalist, was
a contemporary of Linneaus and was often intellectually at odds with him.
There is a rumor that Linneaus, who sometimes named newly discovered plants
and animals after contemporaries, named a toad after Buffon, called Bufo
bufo. Because I'm an admirer of Buffon, I commented many times to my
colleagues in Uppsala about my extreme uneasiness at working in an office
not more than a 10minute walk from Linneaus' gardens!

Carolus Linnaeus (17071778)
Click on the picture to reach the home page of Linneaus'
gardens.

An interesting discussion,
with a simulation, can be found at George Reese' Buffon's
Needle site.
References
Khamis,
H. (1988). A note on Buffon's needle problem. Wright State University,
Department of Mathematics & Statistics Technical Report No. 1988.02.
Solomon,
H. (1978). Geometric Probability. Society for Industrial and Applied
Mathematics. 
