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  "body": "\n## Summary\n\nA Monte Carlo method computes a deterministic quantity approximately by drawing random samples.\nIn its basic form the quantity is the [[expected value|0WQNRCNY3M]] of a [[random variable|0PNRKMRC55]].\nIt is estimated by the average of many independent copies.\nThe [[law of large numbers|0ZVEAYVRC3]] makes the estimate converge.\nThe [[central limit theorem|0EPZC4Z1FC]] shows that the error decreases like $1/\\sqrt{n}$ for $n$ samples, whatever the dimension.\nThis makes Monte Carlo methods the tool of choice for integrals in high dimension and for complex stochastic models.\n\n## Explanations\n\n### L1 human\n\nTo estimate the area of a pond, throw many stones at random into a square field around it and count the fraction that land in the water.\nThat fraction times the area of the field is close to the area of the pond, and closer when more stones are thrown.\nThis is a Monte Carlo method: a number that has nothing random about it is found by repeated random experiments.\nFor example, if $10\\,000$ random points in the unit square give $7\\,850$ points with $x^2 + y^2 \\le 1$, the estimate of $\\pi/4$ is $0.785$, so $\\pi \\approx 3.14$.\nThe price is slow accuracy: to gain one more decimal digit one needs about a hundred times more samples.\n",
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