# Buffon's needle

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In mathematics, **Buffon's needle problem** is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:

- 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?

Using integral geometry, the problem can be solved to get a Monte Carlo method to approximate π.

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## [edit] Solution

The problem in more mathematical terms is: Given a needle of length *l* dropped on a plane ruled with parallel lines *t* units apart, what is the probability that the needle will cross a line?

Let *x* be the distance from the center of the needle to the closest line, let *θ* be the acute angle between the needle and the lines, and let .

The probability density function of *x* between 0 and *t* /2 is

The probability density function of θ between 0 and π/2 is

The two random variables, *x* and *θ*, are independent, so the joint probability density function is the product

The needle crosses a line if

Integrating the joint probability density function gives the probability that the needle will cross a line:

For *n* needles dropped with *h* of the needles crossing lines, the probability is

which can be solved for *π* to get

Now suppose *t* < *l*. In this case, integrating the joint probability density function, we obtain:

where *m*(θ) is the minimum between (*l* / 2)sinθ and *t* / 2.

Thus, performing the above integration, we see that, when *t* < *l*, the probability that the needle will cross a line is

## [edit] Lazzarini's estimate

Mario Lazzarini, an Italian mathematician, performed the Buffon's needle experiment in 1901. Tossing a needle 3408 times, he attained the well-known estimate 355/113 for π, which is a very accurate value, differing from π by no more than 3×10^{−7}. This is an impressive result, but is something of a cheat, as follows.

Lazzarini chose needles whose length was 5/6 of the width of the strips of wood. In this case, the probability that the needles will cross the lines is 5/3π. Thus if one were to drop *n* needles and get *x* crossings, one would estimate π as

- π ≈ 5/3 ·
*n*/*x*

π is very nearly 355/113; in fact, there is no better rational approximation with fewer than 5 digits in the numerator and denominator. So if one had *n* and *x* such that:

- 355/113 = 5/3 ·
*n*/*x*

or equivalently,

*x*= 113*n*/213

one would derive an unexpectedly accurate approximation to π, simply because the fraction 355/113 happens to be so close to the correct value. But this is easily arranged. To do this, one should pick *n* as a multiple of 213, because then 113*n*/213 is an integer; one then drops *n* needles, and hopes for exactly *x* = 113*n*/213 successes.

If one drops 213 needles and happens to get 113 successes, then one can triumphantly report an estimate of π accurate to six decimal places. If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary. Lazzarini performed 3408 = 213 · 16 trials, making it seem likely that this is the strategy he used to obtain his "estimate".

## [edit] See also

## [edit] External links and references

- Buffon's Needle at cut-the-knot
- Math Surprises: Buffon's Noodle at cut-the-knot
- MSTE: Buffon's Needle
- Buffon's Needle Java Applet
- Estimating PI Visualization (Flash)
- Ramaley, J. F. (October 1969). "Buffon's Noodle Problem".
*The American Mathematical Monthly***76**(8): 916–918. doi:. http://links.jstor.org/sici?sici=0002-9890%28196910%2976%3A8%3C916%3ABNP%3E2.0.CO%3B2-9&size=LARGE. - Mathai, A. M. (1999).
*An Introduction to Geometrical Probability*. Gordon & Breach. http://books.google.com.au/books?id=FV6XncZgfcwC. p. 5 - Animations for the Simulation of Buffon's Needle by Yihui Xie using the R package animation
- 3D Physical Animation Java Applet by Jeffrey Ventrella