Buffon's Needle

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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:

$\map \Pr {E \mid \Theta_\theta} = \dfrac {\cos \theta} 1 = \cos \theta$


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

\(\displaystyle \map \Pr E\) \(=\) \(\displaystyle \sum_{\theta \mathop \in \hointr {-\pi / 2} {\pi / 2} } \map \Pr {E \mid \Theta_\theta} \map \Pr {\Theta_\theta}\) Total Probability Theorem
\(\displaystyle \) \(=\) \(\displaystyle \int_{-\pi / 2}^{\pi / 2} \cos \theta \frac {\d \theta} \pi\)
\(\displaystyle \) \(=\) \(\displaystyle \intlimits {\frac 1 \pi \sin\theta} {-\pi / 2} {\pi / 2}\) Primitive of Cosine Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 \pi \paren {1 - \paren {-1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 \pi\)

$\blacksquare$


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