Commuting probability

In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite groups equipped with a suitable probability measure, and can also be generalized to other algebraic structures such as rings. 
Definition.
Let formula_1 be a finite group. We define formula_2 as the averaged number of pairs of elements of formula_1 which commute:
If one considers the uniform distribution on formula_5, formula_2 is the probability that two randomly chosen elements of formula_1 commute. That is why formula_2 is called the commuting probability of formula_1.