Buffon's needle · Monte-Carlo π

Can pure randomness measure π? Drop needles on a ruled floor and count how often they cross a line.

How the lab tests it

Drop N needles of length L on a floor ruled with lines spaced d apart (L < d): each needle's centre-to-line distance and angle are uniform, so the crossing probability integrates to P = 2L/(πd). Tally the crossings and invert: π̂ = 2L/(d·P̂). A separate experiment runs 200 independent estimates at each N to measure the estimator's RMS error.

What it checks

π recovered from a crossing tally (π̂ = 2L/(d·P̂)); and the universal Monte-Carlo convergence rate — the RMS error falling as N^(−1/2) (halving the error costs 4× the samples), the same √N law that makes Monte-Carlo integration slow but dimension-blind