Curiosities · Foundation · 9 min
Estimating π with needles
Random geometry can estimate a constant through a known crossing probability.
WHAT HAPPENS IF…What happens if no needles have crossed yet?
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The midpoint distance to the nearest line is uniform on [0,d/2], independent of uniform orientation. A crossing occurs when that distance is at most (ℓ/2)|sinθ|. Averaging the conditional probability over θ gives 2ℓ/(πd). Solving with an empirical crossing fraction gives a noisy estimate of π.
A detail that matters
The short-needle formula does not apply unchanged when ℓ>d.
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