Condorcet's Jury Theorem shows that on a dichotomous choice, individuals who all have the same competence above 0.5, can make collective decisions under majority rule with a competence that approaches 1 as either the size of the group or the individual competence goes up. The theorem assumes that the probability of each voter's being correct is independent of the probability of any other voter being correct. Contrary to several authors, the presence of mutual or common influences such as opinion leaders does not easily rule independence either in or out. Indeed, and this ought to be surprising,under certain conditions deference to opinion leaders can improve individual competence without violating independence, and so can raise group competence as well.