Solution
Buffon's Needle
Show the problem again
A needle of length ℓ is dropped at random onto a floor ruled with parallel lines spaced ℓ apart. What is the probability the needle crosses a line? Give an exact expression.
Worked solution
The answer is 2/π ≈ 0.6366. Let x be the distance from the needle's center to the nearest line (uniform on [0, ℓ/2]) and θ its angle to the lines (uniform on [0, π/2]). The needle crosses iff x ≤ (ℓ/2)sin θ. Integrating: P = (2/π)·(2/ℓ) ∫₀^{π/2} (ℓ/2)sin θ dθ = 2/π. This gives a Monte Carlo method for estimating π.
Source: Posed by Georges-Louis Leclerc, Comte de Buffon (1777). Statement written for AxiomIQ.