π from Darts

Question

You throw 1,000 darts uniformly at a unit square containing the inscribed quarter-circle of radius 1, and 785 land inside the quarter-circle. What's your estimate of π, and roughly how accurate should you expect it to be?

Accepted Answer

Each dart lands in the quarter-circle with probability fractextarea1 = fracpi4, so hatpi = 4 times frac7851000 = mathbf3.14. Accuracy. The hit fraction has standard error sqrtfracpqn = sqrtfrac0.785 times 0.2151000 approx 0.013, so textSE(hatpi) = 4 times 0.013 approx 0.052. The estimate is 3.14 pm 0.05, you know π to about one decimal place, and the frac1sqrt n law says each extra digit of precision costs 100× more darts (~10 million darts for 3 decimals). Monte Carlo's signature trade-off in one line: the convergence rate O(n^-1/2) is dimension-independent (why it wins for high-dimensional integrals like basket-option pricing) but painfully slow in absolute terms (why no one computes π this way, and why variance-reduction tricks, antithetic draws, control variates, importance sampling, are a real interview topic at simulation-heavy shops).

π from Darts

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