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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.

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 Rose-Hulman 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 1999-2000 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 10-minute walk from Linneaus' gardens!

Carolus Linnaeus (1707-1778)
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.

 Statistical Consulting Center Dept of Mathematics and Statistics College of Science and Math WSU Home

 Prof. Harry Khamis, Director [e-mail] Statistical Consulting Center [home][e-mail] 130 Mathematics and Microbiological Sciences Building Wright State University 3640 Colonel Glenn Highway Dayton OH 45435 USA office phone: (937) 775-2433 or stat consulting center (937) 775-4205 fax: (937) 775-2081 Last updated: July 18, 2001 All Wright State University Webpages must publish this disclaimer. Please send your comments, questions and suggestions regarding this webpage to me. (sjm)