Generalizing Buffon's Needle: An Investigation in Geometric Probability

In 1733, the mathematician Georges-Louis Leclerc Comte de Bu.on stated the following question which is known as Buffon's needle problem:

"Suppose we have a floor made of parallel strips of wood, each the same width t, and we drop a needle of length l onto the floor. What is the probability that the needle will lie across a line between two strips?"

The earliest example of a problem from the currently active field of geometric probability, Buffon solved this problem in 1777. When the length of the needle l is smaller than the width of the parallel strips of wood t, one can use basic geometry to show that the probability P that the dropped needle will cross a line between two strips is given by the following formula

P=(2l)/(tπ).

An interesting observation is that experiments that estimate the theoretical probability P will then also estimate π, although the rate of convergence of the estimates is quite slow.

This quarter, we will study the original Buffon's needle problem and the related probabilistic approach to approximate the value of π.. We will then focus on solving an 'equivalent' problem on the sphere and discuss

the original problem as a limiting case. If time permits, we will discuss possible extensions of the problem to different shapes. Geometric probability is important to the study of both probability and geometry because it involves translating problems from one field to the other and sometimes problems in one field can be answered more easily using the tools developed in the other. It is also a field with great potential for "real

world" applications, for example, in problems involving random networks and computing. This project will open up to students an exciting area of research by introducing them to an early and archetypal question in the field and discussing a non-trivial way of generalizing it.