# Buffon's Needle

## Theorem

Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards.

Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height.

Then the probability that the needle falls across one of the parallel lines is $\dfrac 2 \pi$.

## Proof

For simplicity, consider the real number plane $\R^2$ divided into strips by the lines $x = k$ for each integer $k$.

Then the needle would have length $1$, which is the distance between the lines.

Define $\theta \in \hointr {-\dfrac \pi 2} {\dfrac \pi 2}$ as the angle between the needle and the $x$-axis.

Then the horizontal component of length of the needle is $\cos \theta$ for each $\theta$.

Let:

- $E$ be the event where the needle falls across the vertical lines,
- $\Theta_\theta$ be the event where the angle between the needle and the $x$-axis is $\theta$.

Let the needle drop.

Without loss of generality, let the end with the larger $x$-coordinate have $x$-coordinate $0 \le x_n < 1$.

Then for each $\theta$, the needle falls across the line $x = 0$ exactly when $0 \le x_n \le \cos \theta$.

Therefore the probability that this happens is:

- $\condprob E {\Theta_\theta} = \dfrac {\cos \theta} 1 = \cos \theta$

By considering $\theta$ as a continuous random variable,

\(\ds \map \Pr E\) | \(=\) | \(\ds \sum_{\theta \mathop \in \hointr {-\pi / 2} {\pi / 2} } \condprob E {\Theta_\theta} \map \Pr {\Theta_\theta}\) | Total Probability Theorem | |||||||||||

\(\ds \) | \(=\) | \(\ds \int_{-\pi / 2}^{\pi / 2} \cos \theta \frac {\d \theta} \pi\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \intlimits {\frac 1 \pi \sin\theta} {-\pi / 2} {\pi / 2}\) | Primitive of Cosine Function | |||||||||||

\(\ds \) | \(=\) | \(\ds \frac 1 \pi \paren {1 - \paren {-1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac 2 \pi\) |

$\blacksquare$

## Also known as

This problem is also known just as the **needle problem**.

## Source of Name

This entry was named for Georges Louis Leclerc, Comte de Buffon.

## Historical Note

Georges Louis Leclerc, Comte de Buffon published this problem in his *Histoire Naturelle* in $1777$.

Pierre-Simon de Laplace extended the problem to a general rectangular grid, thus creating what is now sometimes referred to as the Buffon-Laplace Problem.

Augustus De Morgan reports that a pupil of his once performed a practical experiment using Buffon's Needle to calculate a value for $\pi$.

After $600$ trials, a value of $3.137$ was obtained.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Buffon's needle problem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Buffon's needle problem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**needle problem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Buffon's needle**