Probability and Combinatorial Group Theory. If this is too broad or is otherwise a poor question, I apologise.
I learnt recently that the probability that two integers generate the additive group of integers is $\frac{6}{\pi^2}$.

What other results are there like this?

I'm looking for any results of probability applied to group theory, preferably combinatorial group theory, in manner such as the one above.

For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes. 
