Buffon’s needle problem is named after Georges-Louis Leclerc, Comte de Buffon, who lived in the 18 th century. That problem solved by Buffon was the earliest geometric probability problem to be solved.
Theorem (Buffon’s needle problem, problem first posed in 1733, solved in 1777). (Georges-Louis Leclerc, Comte de Buffon)
The probability that a needle of length will randomly land on a line, given a floor with equally spaced parallel lines at a distance apart, is .
Proof. (assuming that the angle and the position of the fallen needle are independently and uniformly random)
If the needle always fell perpendicular (angle radians) to the parallel lines, we would have . So we have
|P (l, d ) = sin θ ⋅ ⋅ P⊥(l, d ) = ⋅ = ⋅ .|