All nonabelian groups are nonabelian, but some nonabelian groups are more nonabelian than others. One way to make this idea precise is through the commuting probability of a group, that is, the probability that two randomly chosen group elements commute. As it turns out, this quantity encodes a surprising amount of information about the group! In this talk, I'll introduce the commuting probability and illustrate its properties with several examples. I'll also discuss how these ideas lead to the notion of isoclinism of groups, which is a useful organizational tool appearing in classification problems in finite group theory throughout history. No background is necessary aside from basic group theory (centers, centralizers, etc.) and familiarity with the number 5/8.