Buffon's Needle

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Theorem

Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards.

Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height.


Then the probability that the needle falls across one of the parallel lines is $\dfrac 2 \pi$.


Proof


Source of Name

This entry was named for Georges Louis Leclerc, Comte de Buffon.


Historical Note

Georges Louis Leclerc, Comte de Buffon published this problem in his Histoire Naturelle in $1777$.

Pierre-Simon de Laplace extended the problem to a general rectangular grid, thus creating what is now sometimes referred to as the Buffon-Laplace Problem.

Augustus De Morgan reports that a pupil of his once performed a practical experiment using Buffon's Needle to calculate a value for $\pi$.

After $600$ trials, a value of $3.137$ was obtained.


Sources