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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 pi.
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 t > l, x be the distance from the center of the needle to the closest line, and θ be
the acute angle between the needle and the lines.
The probability density function of
x between 0 and t/2 is
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The probability density function of
θ between 0 and π/2 is
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The two random variables, x and θ, are
independent, so the joint probability density function is the product
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The needle crosses a line if
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Integrating the joint probability density function gives the probability that the needle will cross a line:
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For n needles dropped with h of the needles crossing lines, the probability is
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which can be solved for π to get
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