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## Probability of seeing a stage, part II

In the previous post in this series, I described a problem dealing with a lattice with an origin, and asked what is the probability that one can see the origin from a random place in the lattice. In this post I’ll give you my *new* proof of this result, with the understanding that someone else might have done this same proof before me (I’m not aware of such person so I will claim discovery).

As Carl Brannen pointed out in the comments, this is the same as the probability that when one picks two random integers they are relatively prime. I’ll leave that to you as a proof. This is a famous result in mathematics: it is given by

$\zeta(2)^{-1} = \frac{6}{\pi^2}$

It is also sometimes stated as “Most fractions are reduced”, seeing as fractions involve the ratio of two integers.

The first time I heard about this result I was about 15 years old. It was told to me without proof and I thought it must be really hard to proof, because how does one pull all those factors of $\pi$ by counting?

Here below is an illustration of the situation.

A region near the origin.

I’ve graphed a region of the plane near the origin. If you don’t pay too much attention to the details, you will notice that the figure is rather uniform in color. We have to show that this uniformity of color persists to a large enough (infinite) size.