What Is the Probability of Seeing a Point on an Infinite Grid?

[h6ss]

Here is a difficult probability question I found interesting and thought I'd share:

Suppose you are standing on an infinitely large square grid at the point (0,0), and suppose that you can see infinitely far but cannot see through grid points. Given a random grid point z = (x, y), where x and y are integers, what is the chance you can see z?

The rather elegant answer is:

Spoiler

Prob = \frac{6}{\pi^2}

More information:

 

[mfb]

There is no uniform distribution over all integers. What does "random integer" mean?
The answer does not depend much on it, but there are choices that do give a different answer.

I think the problem is not as hard as the pdf describes it, as the steps taken in the first solution are not hard to find and the value of the product is well-known.

 

[Astudious]

What is the question actually asking? "What is the probability that the line connecting a randomly assigned point z (x,y) with the origin does not intersect any grid points"?