Notes on Buffon's Experiments
Notes on Buffon's Experiments |
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Simulation of Buffon's coin experiment
Simulation of Buffon's needle experiment
Georges-Louis Leclerc, Compte de Buffon was born in 1707 in Montbard, France. He was the director of the Paris Jardin du Roi and was best known during his time for his thirty-six volume work on natural history. The coin and needle problems are considered to be among the first problems in geometric probability and greatly stimulated the development of the subject. Buffon died in 1788. For more historical information, visit the History of Mathematics web site.
The original needle problem has been extended in many ways, starting with Laplace who considered a floor with rectangular tiles. Indeed, variations on the problem are active research problems even today. For a thorough mathematical treatment of the needle problem and its many extensions, see the book Geometric Probability.
If you are ever in Montbard, please visit the Buffon Museum. The address is
Musée Buffon
BP 90
21506 Montbard Cedex
Fax (33).80.89.11.99
It is very easy to simulate the basic outcome variables in Buffon's experiments using random numbers. The key idea is given in the following problem:
1. Suppose that the random variable U is uniformly distributed on the
interval (0, 1) (that is, U is a random number). Let a
and b be real numbers with a < b and
define random variable W by
W = a + (b - a)U
Show that W is uniformly distributed on the interval (a, b).
2. Let X and Y
denote the coordinates of the center of the coin in Buffon's coin
experiment. Use the result of Exercise 1 to express X
and Y in terms of random numbers.
3. Let X and Y
denote the angle and distance variables in Buffon's needle
experiment. Use the result of Exercise 1 to express X
and Y in terms of random numbers.
Neil Weiss has pointed out that our computer simulation of Buffon's needle experiment is circular, in the sense the program assumes knowledge of pi (you can see this from the result of Exercise 3).
4. Try to write a computer algorithm
for Buffon's needle problem, without assuming the value of pi or
any other transcendental numbers.
Buffon's Experiments |
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