If you pick $\displaystyle p,q \in \mathbb{N}$, what is the probability that $\displaystyle p$ and $\displaystyle q$ are relatively prime?

Does this probability even exist?

Since my guess is that it doesn't exist, what if you pick $\displaystyle p,q \in \mathbb{N}$ such that $\displaystyle p < n, q < n$? Can you calculate the probability that $\displaystyle p$ and $\displaystyle q$ are relatively prime if there is an upper bound?