In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:

Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is

This can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question. The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.

The problem in more mathematical terms is: Given a needle of length l dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will lie across a line upon landing?

Let x be the distance from the center of the needle to the closest parallel line, and let θ be the acute angle between the needle and one of the parallel lines.

The uniform probability density function (PDF) of x between 0 and ⁠t/2⁠ is

Here, x = 0 represents a needle that is centered directly on a line, and x = ⁠t/2⁠ represents a needle that is perfectly centered between two lines. The uniform PDF assumes the needle is equally likely to fall anywhere in this range, but could not fall outside of it.

The uniform probability density function of θ between 0 and ⁠π/2⁠ is