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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:
## Solution

## Lazzarini's estimate

## See also

## External links and references

- 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.

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 $tge\; l$.

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

- $frac\{2\}\{t\},dx.$

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

- $frac\{2\}\{pi\},dtheta.$

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

- $frac\{4\}\{tpi\},dx,dtheta.$

The needle crosses a line if

- $x\; le\; frac\{l\}\{2\}sintheta.$

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

- $int\_\{theta=0\}^\{frac\{pi\}\{2\}\}\; int\_\{x=0\}^\{(l/2)sintheta\}\; frac\{4\}\{tpi\},dx,dtheta\; =\; frac\{2\; l\}\{tpi\}.$

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

- $frac\{h\}\{n\}\; =\; frac\{2\; l\}\{tpi\},$

which can be solved for π to get

- $pi\; =\; frac\{2\{l\}n\}\{th\}.$

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

- $int\_\{theta=0\}^\{frac\{pi\}\{2\}\}\; int\_\{x=0\}^\{m(theta)\}\; frac\{4\}\{tpi\},dx,dtheta\; ,$

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

- $frac\{h\}\{n\}\; =\; frac\{2\; l\}\{tpi\}\; -\; frac\{2\}\{tpi\}left\{sqrt\{l^2\; -\; t^2\}\; +\; tsin^\{-1\}left(frac\{t\}\{l\}right)right\}+1.$

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 = 113n/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 113n/213 is an integer; one then drops n needles, and hopes for exactly x = 113n/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".

- 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. "Buffon's Noodle Problem".
*The American Mathematical Monthly*76 (8): 916–918. - Mathai, A. M. An Introduction to Geometrical Probability. Gordon & Breach. p. 5

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Last updated on Tuesday September 09, 2008 at 18:08:16 PDT (GMT -0700)

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Last updated on Tuesday September 09, 2008 at 18:08:16 PDT (GMT -0700)

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