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**TriKri** That random number thing is interesting, cause I don't understand how $\displaystyle \pi$ can be in this kind of discrete probability, if I may say so. I guess that two randomly chosen positive numbers a and b both in the interval $\displaystyle [1, \infty)$ has this chance of not sharing any prime factor: $\displaystyle \prod_{p\text{ prime}} 1 - \frac{1}{p^2}$ because the chance is $\displaystyle \frac{1}{p}$ for a to be divisible by p and hence have prime factor p, and the same is for b. For both a and b to have prime factor p the chance is $\displaystyle \frac{1}{p^2}$. For a and b to not share prime factor p the chance is $\displaystyle 1 - \frac{1}{p^2}$. And for a and b to not share any of the primes as a prime factor the chance is $\displaystyle \prod_{p\text{ prime}} 1 - \frac{1}{p^2}$. Though I have no idea how to connect that formula with $\displaystyle \pi$.