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The above "answer" should probably be a comment elsewhere. In any case, it's worth noting that while $Pr(G) \leq 5/8$ when $G$ is a group (and there are various other values below $5/8$ that are not the commuting probability for any group $G$) the commuting probability of a semigroup can be anything. This is proved <a href="www-rohan.sdsu.edu/~vadim/ps.pdf">here</a>.
