We extend Aumann's [3] theorem, deriving correlated equilibria as a consequence of common priorsand common knowledge of rationality, by explicitly allowing for non-rational behavior. We replacethe assumption of common knowledge of rationality with a substantially weaker one, joint p-belief ofrationality, where agents believe the other agents are rational with probabilities p = (pi)i?I or more.We show that behavior in this case constitutes a type of correlated equilibrium satisfying certain p-beliefconstraints, and that it varies continuously in the parameters p and, for p sufficiently close to one, withhigh probability is supported on strategies that survive the iterated elimination of strictly dominatedstrategies. Finally, we extend the analysis to characterizing rational expectations of interim types, togames of incomplete information, as well as to the case of non-common priors.