---
tag: 06X5PF6YFP
kind: problem
name: Buffon's needle problem
aliases: [buffons-needle]
msc: [60D05]
uses: [0924BCD257]
sources:
  - url: https://en.wikipedia.org/wiki/Buffon's_needle_problem
    locator: Solution
same_as:
  wikidata: Q1753097
  wikipedia: Buffon's needle problem
---

## Problem

A floor has parallel lines at distance $t > 0$ apart. A needle of length $l$ with $0 < l \le t$ falls at random on the floor.
"At random" means this.
The distance $x$ of the centre of the needle from the nearest line is uniform on $[0, t/2]$.
The acute angle $\theta$ between the needle and the lines is uniform on $[0, \pi/2]$.
The variables $x$ and $\theta$ are independent.
Find the probability that the needle crosses a line.

## Remarks

The answer is $\dfrac{2l}{\pi t}$. The needle crosses a line exactly when $x \le \frac{l}{2} \sin \theta$, and the integral of this region gives the answer.
For a needle with $l = t$, the probability is $2/\pi \approx 0.637$.
The answer contains $\pi$, so repeated throws give a random estimate of $\pi$.
For a long needle with $l > t$, the answer is a different formula.

## History

Georges-Louis Leclerc, Comte de Buffon, posed the problem in the 18th century. It was one of the first problems of geometric probability.

## Explanations

### L1 human

The needle crosses a line when it lands close to a line and at a steep angle.
The chance is the area of the "good" pairs (distance, angle) divided by the area of all pairs.
A sine function appears in this area. That is how $\pi$ comes into the answer.