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.

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

@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.

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