Statistics

Monte Carlo

Throw random darts at a box and count the ones that land in a shape: the fraction times the box is its area. Estimate π, the area under a curve or the volume of a ball in ten dimensions, and watch the error fall like 1/√n.

hitmisserror |estimate − true|expected error ∝ 1/√n

Estimate of π—

—

Estimate

Darts per frame 100

Numbers

darts thrown n—

hits—

true value—

error now—

expected error A√(p(1−p)/n)—

What to observe

Run the π estimate. The error jitters but hugs the dashed1/√n line: a hundred times more darts buys only one more digit.
Area under sin x is the same recipe: box area times hit fraction. Any shape you can test a point against has an area this way.
Push the ball to d = 10. It fills a quarter of a percent of the cube, so hits are rare and the error is large next to the answer, yet it still falls like 1/√n in any dimension.

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