Concept
Solving hard problems by random simulation — from atomic bombs to modern AI.
Understand it in one breath
"Approximate hard integrals or probabilities with random samples, then analyze the error." Ulam, von Neumann, Metropolis, and colleagues developed the modern method collaboratively at Los Alamos. Pi estimation, option pricing, particle simulation, and some tree searches share sampling ideas but use different algorithms and guarantees.
At a glance
Samples N | Estimate of π | Error |
|---|---|---|
100 | ~3.08 | ~0.06 |
1,000 | ~3.156 | ~0.014 |
10,000 | ~3.1432 | ~0.0016 |
100,000 | ~3.1416 | ~0.0001 |
1,000,000 | ~3.14159 | ~10⁻⁵ |
Throw random points into the square [−1,1]²; the fraction that land inside the circle × 4 ≈ π. Error is inversely proportional to √N — four times the precision requires sixteen times as many samples.
Key formula
Key moments
Ulam — inspiration from a card game
While recovering from illness, Stanisław Ulam considered estimating the chance of winning solitaire through repeated random trials rather than exhaustive calculation.
von Neumann brings the method to ENIAC
Ulam, von Neumann, and colleagues applied random sampling to neutron-transport calculations. The name Monte Carlo evoked the famous casino and games of chance.
AlphaGo and Monte Carlo tree search
Monte Carlo tree search, combined with neural networks, helped AlphaGo defeat Go champion Lee Sedol and reshaped AI game-playing strategy.
Modern applications
Financial option pricing, particle and molecular simulation, ray tracing in film, and reinforcement-learning simulators.
Beyond MathVoyage
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