Buffon's needle problem

In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:^[1]

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

Buffon's needle was the earliest problem in geometric probability to be solved;^[2] 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

${\displaystyle p={\frac {2}{\pi }}\cdot {\frac {l}{t}}.}$

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.^[3] The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.