Problem. Find an upper bound on the maximum probability that two randomly chosen elements of a finite non-abelian group commute, and show that it is attained.
This gives us two things to compute (or more correctly, to bound): the probability that a random element commutes with everything, and the probability that if it does not commute with everything, then it commutes with a random element. The natural approach is then clearly thinking about centers and centralisers.
Let $G$ be a finite group. If it is non-abelian, then its center $Z(G)$ is not the entire group, that is, $G$ has some element $g \not \in Z(G)$. Then $C_G(g)$, centraliser of $g$, also cannot be the entire group, as that would contradict $z \not \in Z(G)$.