Buffon-Laplace Problem

Theorem

Let a horizontal plane be divided into rectangles by a rectangular grid of straight lines of distances $a$ and $b$ apart.

Let a needle whose length equals $l$ be dropped onto the plane randomly from a random height.

Then the probability that the needle falls across one or more lines is $\dfrac {2 l \paren {a + b} - l^2} {\pi a b}$.

Proof

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Also known as

The problem of Buffon's Needle, along with the Buffon-Laplace Problem, is also known just as the Needle Problem.

Source of Name

This entry was named for Georges Louis Leclerc, Comte de Buffon and Pierre-Simon de Laplace.

Historical Note

Pierre-Simon de Laplace presented this extension to the problem of Buffon's Needle in $1812$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Buffon's needle problem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Buffon's needle problem