Estimating pi by throwing darts

Question

A circle of radius 1 is drawn inside a square of side 2, touching all four sides. Points are thrown at random, uniformly over the square. 1. How does the fraction of points landing inside the circle give an estimate of pi? 2. Roughly how many points are needed to get pi correct to two decimal places?

Accepted Answer

The fraction inside the circle is close to pi/4, so pi is about 4 times that fraction. For two decimal places you need on the order of a hundred thousand points. The ratio of areas. The square has area 4. The circle has area pi × 1 squared = pi. A uniformly random point lands inside the circle with probability pi/4, about 0.785. So if N points are thrown and k land inside, k/N is close to pi/4 and 4k/N estimates pi. How many points. The fraction k/N has standard deviation about the square root of p(1 minus p)/N with p = 0.785, which is about 0.41 / square root of N. The estimate 4k/N therefor…

Estimating pi by throwing darts

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