Buffon problem

of the needle

A classical problem in the theory of geometric probabilities, which is rightly considered to be the starting point in the development of this theory. First posed by G. Buffon in 1733 and reproduced together with its solution in [1]. Buffon considered the following situation: A needle of length $2r$, where $2r<a$, is thrown at random on a plane ruled by parallel lines at a distance $a$ from one another. What is the probability that the needle thus thrown will come to rest across one of the lines? Clearly, the location of the needle is defined by the distance $x$ from its centre to the nearest straight line and by the acute angle $\theta$ between the needle and the perpendicular to the line. The magnitude $x$ lies between zero and $a/2$, while $\theta$ lies between zero and $\pi/2$. It is assumed that the point $(x,\theta)$ is uniformly distributed in the appropriate rectangle (this is equivalent to assuming that the random variables $x$ and $\theta$ are independent and are uniformly distributed over $(0,a/2)$ and $(0,\pi/2)$). Then the sought probability is defined as the ratio between the areas corresponding to the favourable and to all possible results, and is

\begin{equation}p=\frac{1}{\frac a2\frac\pi2}\int\limits_0^{\pi/2}r\cos\theta d\theta=\frac{4r}{a\pi}.\label{*}\end{equation}

In the past Buffon's problem served as the tool for the experimental check on the Bernoulli theorem. In fact, if the needle is thrown $n$ times, and crosses one of the lines in $m$ cases, the frequency $m/n$ should, for large values of $n$, according to Bernoulli's theorem, be close to the probability \eqref{*}. This idea was used by many workers for the determination of the number $\pi$ by random trials [1], [2]. Buffon also considered other similar problems, in particular the problem of the needle crossing the lines belonging to two mutually-perpendicular systems which subdivide the plane into rectangles having sides of lengths $a$ and $b$ respectively. The answer given by Buffon to this problem is incorrect. The correct solution

$$\frac{4r(a+b)-4r^2}{\pi ab}$$

was found by P. Laplace in 1812.

References

[1]	G. Buffon, "Essai d'arithmétique morale. Supplement à "l'Histoire Morale" " , 4 (1777)
[2]	J.V. Uspensky, "Introduction to mathematical probability" , McGraw-Hill (1937)
[3]	M.G. Kendall, P.A.P. Moran, "Geometric probability" , Griffin (1963)