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

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 $3.167$ was obtained.


Sources