Buffon’s Needle Problem

In 1733, Georges-Louis Leclerc, Comte de Buffon, posed the question: what is the probability that a needle, when dropped randomly on a ruled surface, will intersect one of the lines? The solution, published in 1777, is a classic result in geometric probability. To solve it, let the needle have length \(l\) and the parallel lines be separated by a distance \(t \ge l\). The position of the needle can be described by two variables: the distance \(x\) from the center of the needle to the nearest line, and the angle \(\theta\) the needle makes with the lines. The variable \(x\) is uniformly distributed in \([0, t/2]\), and \(\theta\) is uniformly distributed in \([0, \pi/2]\).

The needle crosses a line if \(x \le \frac{l}{2}\sin\theta\). The problem is to find the area of this region in the \((x, \theta)\) parameter space and divide it by the total area of the parameter space, which is \(\frac{t}{2} \times \frac{\pi}{2} = \frac{\pi t}{4}\). The area of the “favorable” region (where a crossing occurs) is given by the integral \(\int_0^{\pi/2} \frac{l}{2}\sin\theta \,d\theta = \frac{l}{2}[-\cos\theta]_0^{\pi/2} = \frac{l}{2}\). The probability is the ratio of these areas: \(P = \frac{l/2}{\pi t/4} = \frac{2l}{\pi t}\). By performing the experiment many times and observing the frequency of crossings, one can rearrange the formula to estimate \(\pi\): \(\pi \approx \frac{2l}{tP}\). This physical simulation to solve a mathematical problem is a direct intellectual ancestor of modern Monte Carlo methods.