Estimating π (Monte Carlo)
Random dice, a fixed ratio
Summary
Throw random points into a unit square. The fraction that land inside the quarter circle approaches π/4. You can calculate π just by rolling dice: a statistical miracle.
In depth
The Monte Carlo method solves hard mathematical problems by random sampling. Estimating π is its simplest example. Random points (x, y) between 0 and 1 fill the unit square evenly. The area of the quarter circle of radius 1 is π/4, and the area of the square is 1. Throw enough points and the ratio inside the circle / total approaches π/4. The accuracy of the method improves as 1/√n, so 100 times the accuracy needs 10,000 times as many points. It looks slow, but for high-dimensional integrals (for example in physics simulations or the pricing of financial derivatives) it is far more efficient than classical methods.
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