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BellaJ

Is the coconut experiment evaluating PI based on how many times the coconut lands inside the circle vs outside the circle ?

jzhanson

@BellaJ That's right! If we let the square have side length of 1, then it has area 1. The circle, inscribed in the square, has radius 0.5 and area $\pi r^2 = \pi (0.5)^2 = \frac{\pi}{4}$.

Then, we have

$$\frac{\text{Area of the circle}}{\text{Area of the square}} = \frac{\pi / 4}{1} = \frac{\pi}{4}$$

$$\implies \frac{\pi}{4} = \frac{\text{Coconuts in circle}}{\text{All coconuts}} \implies \pi \approx 4 \times \frac{\text{Coconuts in circle}}{\text{All coconuts}}$$

And, as we throw more and more coconuts, the ratio of coconuts in the circle to all coconuts thrown gets closer and closer to the true value of $\pi$, as desired.

HelloWorld

Just a fun fact: some distributions have infinite variance which breaks some of these guarantees.