# 37% chance

I don’t know if you ever realized, before, but it is quite common to have 37% chance that something happened (or actually “not happened” if we want to be more rigorous). For instance, consider a $n\times n$ grid, and draw $n^2$ points randomly (and uniformely). Then, around 37% cells are empty. Or if you consider a cell, on that grid, there is 37% chance, that the cell is empty. You can look, on the animation below,

Actually, it is quite simple to prove this result. This come from the fact that

$\exp(-1)=0.36787944117\sim 0.37$

And $\exp(-1)$ is a common probability. To be more specific, we can write it

$\exp(-1)\frac{1^0}{0!}$

which is $\mathbb{P}(N=0)$ when $N$ is a Poisson distribution with parameter $\lambda=1$ (or with mean $1$). In our case, $N$ is the number of points in a given cell, and since there are $n^2$  points for $n^2$  cells (it is a $n\times n$ grid), then $\mathbb{E}(N)=1$. And the Poisson distribution arise here as a limit of a binomial distribution. More precisely, here $\mathcal{B}(n^2,1/n^2)$. We have here the law of small number. Nice isn’t it?

## 2 thoughts on “37% chance”

1. Simone Scardapane says:

Really interesting, I never considered that!

2. pela says:

Superbe, merci, je l’integre dans mes cours des la rentrée.

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