Buffon's needle problem - Wikipedia

Abstraction: 18th-century geometric probability problem yielding pi via needle drops

Key points:

• For a needle of length l dropped on parallel lines t apart (l <= t): P(crossing) = 2l/(t*pi); can be rearranged to estimate pi
• For the long needle case (l > t): P(crossing) = 2l/(tpi) - (2/pi)(sqrt(l^2-t^2)/t - arccos(t/l))
• Laplace's extension to perpendicular grid: P = 2l(a+b)/(piab) - l^2/(piab); using two sets of lines is more statistically efficient than one set
• Mario Lazzarini (1901) tossed 3,408 needles to obtain 355/113 approximation of pi, likely by choosing n as multiples of 213; suspected confirmation bias or fabricated data
• Equivalent elegant proof by Barbier (1860): no integration needed, uses the fact a circle of diameter t always crosses lines exactly twice
• Simplified realization by Szyszkowicz (2024) selects random points on a circle to simulate needle rotation without a priori knowledge of pi

Connections: Buffon · Geometric Probability · Monte Carlo Methods · Pi Estimation

Source: http://en.wikipedia.org/wiki/Buffon's_needle