Buffon's needle

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.

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 $$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},d heta.

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

:$$frac{4}{tpi},dx,d heta.

The needle crosses a line if

:$$x le frac{l}{2}sin heta.

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

:$$int_{ heta=0}^{frac{pi}{2 int_{x=0}^{(l/2)sin heta} frac{4}{tpi},dx,d heta = 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:

:$$int_{ heta=0}^{frac{pi}{2 int_{x=0}^{m( heta)} frac{4}{tpi},dx,d heta ,where $$m( heta) is the minimum between$$(l/2)sin heta and $$t/2 .

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} ight) ight}+1.

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&times;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

:&pi; ≈ 5/3 &middot; "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 &middot; "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".

See also

* Buffon's noodle

External links and references

* [http://www.cut-the-knot.org/fta/Buffon/buffon9.shtml Buffon's Needle] at cut-the-knot
* [http://www.cut-the-knot.org/ctk/August2001.shtml Math Surprises: Buffon's Noodle] at cut-the-knot
* [http://www.mste.uiuc.edu/reese/buffon/buffon.html MSTE: Buffon's Needle]
* [http://www.angelfire.com/wa/hurben/buff.html Buffon's Needle Java Applet]
* [http://www.metablake.com/pi.swf Estimating PI Visualization (Flash)]
*
* p. 5