Buffon's needle problem

This article page is a stub, please help by expanding it.

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

P (l, d )

that a needle of length

l

will randomly land on a line, given a floor with equally spaced parallel lines at a distance

d   ≥   1

apart, is

P (l, d ) = 2 π  ⋅   l d

.

Proof. (assuming that the angle and the position of the fallen needle are independently and uniformly random) If the needle always fell perpendicular (angle

θ = π 2

radians) to the parallel lines, we would have

P⊥(l, d ) = l d

. So we have

P (l, d ) = ∫ π 0 sin θ   ⋅   d θ π   ⋅  P⊥(l, d ) = − [cos θ  ] π0 π  ⋅   l d = 2 π  ⋅   l d  .

 □

See also

• Buffon’s constant
• Buffon’s noodle problem