This is mostly joint work with Motiejus Valiunas but concerns a range of questions that I have thought about for a little time and will also mention the work of Yago Antolin, Matthew Tointon and Enric Ventura.

This will be a recap of various results and possible methods for assigning a probability to a group, measuring the event that two elements commute. I will quickly recap a classical result from finite groups, then move on to residually finite groups and amenable groups before outlining a new method for calculating this probability via what we call "coset correct means". A coset correct mean is a function that assigns to each subset of a group a number between 0 and 1, which is finitely additive, assigns 1 to the whole group and gives the "correct" answer for any coset of any subgroup (that number being the reciprocal of the index of the subgroup). I will outline the construction of these coset correct means.

The main result in these contexts is that the probability that two elements of a group commute is non-zero exactly when the group itself is FAF - finite-by-abelian-by-finite. In many situations - such as when the group is finitely generated - a group is FAF just means that it is virtually abelian. That is a finitely generated FAF group is one that has an abelian subgroup of finite index.