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  grid, and draw  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

And  is a common probability. To be more specific, we can write it

which is  when  is a Poisson distribution with parameter  (or with mean ). In our case,  is the number of points in a given cell, and since there are   points for   cells (it is a  grid), then . And the Poisson distribution arise here as a limit of a binomial distribution. More precisely, here . We have here the law of small number. Nice isn’t it?

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