Buffon was no Buffoon

The Buffon Needle method of estimating ${\pi}$ is hopelessly inefficient. With one million throws of the needle we might expect to get an approximation accurate to about three digits. The idea is more of philosophical than of practical interest. Buffon never envisaged it as a means of computing ${\pi}$ .

Image drawn with Mathematica package in: Siniksaran, Erin, 2008: Throwing Buffon’s Needle [Reference below].

Buffon and his Sticks

One of the earliest problems in geometric probability was posed and solved by Georges-Louis Leclerc, Comte de Buffon in 1733. He considered the question of dropping a stick of length ${\ell}$ onto a wooden floor with floorboards of width ${d > \ell}$ . Two outcomes are possible: the stick may land on a single board, or it may cross the gap between two boards (see figure above).

Buffon regarded this process as a game of chance and he was interested in the odds of a given drop crossing a line. Since the game can be played on a table-top with a needle and a ruled sheet of paper, it is generally known as Buffon’s Needle.

For simplicity, we may assume that ${\ell = d = 1}$ ; this assumption is not limiting. If the stick is dropped ‘at random’, the probability that it crosses a line between boards is ${p = 2/\pi}$ . This is simple to calculate, as we now show.

Let ${y\in(-0.5,+0.5)}$ be the (algebraic) distance from the centre of the stick to the nearest line and ${\theta \in [0,\pi)}$ the angle between stick and line. Then there is a crossing if ${|y|<\frac{1}{2}\sin\theta}$ . In the figure below, the region of the ${(\theta,y)}$ -plane where a crossing occurs is shaded:

In the ${(\theta,y)}$ -plane, the shaded region corresponds to crossings of the needle.

The shaded region has area

$\displaystyle 2\int_0^\pi \frac{1}{2}\sin\theta\,d\theta = [-\cos\theta]_{0}^{\pi} = 2.$

The area of the accessible rectangle is ${\pi}$ . Thus, assuming that ${\theta}$ and ${y}$ are uniformly distributed, the probability of landing in the shaded region is

$\displaystyle p = \frac{2}{\pi} \,.$

Estimating ${\pi}$

Now if we drop a stick a large number of times ${n}$ , and get ${k}$ crossings, we can estimate the probability p by means of the frequency ratio ${k/n}$ , so

$\displaystyle \frac{2}{\pi} \approx \frac{k}{n} \qquad\mbox{or}\qquad \pi \approx \frac{2n}{k} \, .$

But there is a big problem: For a large number of trials, the standard error is

$\displaystyle \epsilon = \sqrt{ \frac{p(1-p)}{n} } \,.$

(See Bent Coins: What are the Odds?). We are concerned only with orders of magnitude, so we write

$\displaystyle n \sim \frac{1}{\epsilon^2} \,.$

If we want a result accurate to three significant figures, we might let ${\epsilon = 10^{-3}}$ , but this implies ${n \sim 10^6}$ . We need something like one million throws to get ${\pi}$ to three significant digits — ignoring factors of order ${O(1)}$ . For six-digit accuracy, we need a trillion throws.

A Mathematica package has been written by Siniksaran (2008) to perform Monte-Carlo experiments, generating random needle tosses and computing estimates of ${\pi}$ . The figure below shows simulations using this package for ${n\in\{100,1000,10000\}}$ .

Simulated Monte Carlo trials of the Buffon needle experiment with (from left to right) ${n\in\{100,1000,10000\}}$ .

The Table below shows estimates and percentage errors computed for ${n\in\{10,100,1000,10000,100000\}}$ . It is clear that the accuracy is increasing with ${n}$ , but painfully slowly. One million terms yield only three significant figures accuracy.

Some History

Why did Buffon propose such an impractical method of estimating ${\pi}$ ? He did not!

In a recent article, Behrends (2014) reaches the following conclusions:

There is no evidence that Buffon did the ‘experiment’. He was interested in the odds of a game of chance, assuming ${\pi}$ to be known.
Buffon did do some experiments on the St. Petersburg paradox, in which a fair coin is tossed repeatedly until the first head shows. With certain pay-outs, the expected gain is infinite.
Laplace realised that Buffon’s experiment could, in principle, be used to estimate ${\pi}$ , but there is no indication that he tried this.
Many experiments with Buffon’s Needle are documented. In one, Lazzerini claimed to have obtained the value ${355/113}$ for ${\pi}$ . This is accurate to seven figures but, as only 3408 throws were involved, it cannot be given credence (see also Badger, 1994).

Many variations on Buffon’s Needle have been proposed. One especially curious result is called Buffon’s Noodle. If the needle is bend into any (plane, rectifiable) curve, the probability of crossing a line remains unchanged. This perplexing result is less incredible when it is realised that multiple crossings for a single throw are now possible, and must be counted accordingly (Ramaley, 1969). But it remains a surprising result.

Sources

${\bullet}$ Badger, Lee, 1994: Lazzarini’s lucky approximation of ${\pi}$ . Mathematics Magazine (Mathematical Association of America), 67, (2), 83-91. doi:10.2307/2690682.

${\bullet}$ Behrends, Ehrhard, 2014: Buffon: Hat er Stoeckchen geworfen oder hat er nicht? Retrospektive, MDMV 22/2014, 50–52. DOI 10.1515/dmvm-2014-0022.

${\bullet}$ Buffon, 1733: Contribution to the Royal Academy of Sciences. From the reports of the Academy, 1733, Pp 43-45. http://gallica.bnf.fr/ark:/12148/bpt6k3530m/f51.image

${\bullet}$ Ramaley, J.~F., 1969: Buffon’s Noodle Problem, American Mathematical Monthly, 76, 916-918.

${\bullet}$ Siniksaran, Erin, 2008: Throwing Buffon’s Needle with Mathematica. The Mathematica Journal, 11.1, 71-90.

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