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Yes, as Arturo says, you probably want what is known as the "commuting probabilty of $G$", cp(G). Bob Guralnick and I proved (among other things) in a Journal of Algebra paper (circa 2006) (without using the classification of finite simple groups) that $cp(G) \to 0$ as $[G:F(G)] \to \infty,$ where $F(G)$ is the largest nilpotent normal subgroup of a finite group $G,$ though sharper results are possible using the classification.