Estimate pi by Monte Carlo

Question

Estimate \pi by simulation: sample points uniformly in the unit square [0,1]^2 and count how many land inside the quarter circle x^2 + y^2 \le 1. The quarter circle has area \pi/4, so \pi \approx 4 \cdot \frac{\text{inside}}{n} Write the estimator, then answer: how many samples do you need for roughly 3 correct decimal places?

Accepted Answer

Each dart is a Bernoulli trial with success probability p = \pi/4. Sum the hits, scale by 4/n: Using a local random.Random(seed) instead of the global functions makes runs reproducible and thread-safe, a habit interviewers notice. Convergence, the part that actually gets probed. The hit count is Binomial(n, p), so the estimator's standard error is \sigma{\hat\pi} = 4\sqrt{\frac{p(1-p)}{n}} \approx \frac{1.64}{\sqrt{n}}. Monte Carlo error shrinks like O(1/\sqrt{n}): each extra digit of accuracy costs 100	imes the samples. For error \sim 10^{-3} you need n \approx (1640)^2 \approx 2.7 million …

Estimate pi by Monte Carlo

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