π by darts

Throw random darts at a square with a circle inside it. The fraction that land in the circle settles at π/4, so four times that fraction estimates π. The estimate crawls in — every extra decimal costs about a hundred times the darts.

All five methods run at once on the same number of throws. Top: the chosen board, and the race of estimates toward π. Middle: the sampling distribution — thousands of independent short experiments, histogrammed, with the bell curve theory predicts. Bottom: the convergence law — error against number of throws, on log–log axes, where the 1/√N rule becomes a straight line of slope −½.

The maths: a dart lands inside when x² + y² < 1; Buffon's needle crosses a floorboard line with probability 2L/(πD). Read What π actually measures for the law of large numbers behind it, and why this is a lovely explanation and a terrible calculator.