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:
Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution, in the case where the needle length is not greater than the width of the strips, can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question.
The problem in more mathematical terms is: Given a needle of length 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.
The uniform probability density function of x between 0 and t /2 is
The uniform 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
Now there are two cases.
Integrating the joint probability density function gives the probability that the needle will cross a line:
A particularly nice argument for this result can alternatively be given using "Buffon's noodle".
Suppose . In this case, integrating the joint probability density function, we obtain:
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