Consider an ensemble of $k$ individual classifiers whose accuracies are known. Upon receiving a test point, each of the classifiers outputs a predicted label and a confidence in its prediction for this particular test point. In this paper, we address the question of whether we can determine the accuracy of the ensemble. Surprisingly, even when classifiers are combined in the statistically optimal way in this setting, the accuracy of the resulting ensemble classifier cannot be computed from the accuracies of the individual classifiers-as would be the case in the standard setting of confidence weighted majority voting. We prove tight upper and lower bounds on the ensemble accuracy. We explicitly construct the individual classifiers that attain the upper and lower bounds: specialists and generalists. Our theoretical results have very practical consequences: (1) If we use ensemble methods and have the choice to construct our individual (independent) classifiers from scratch, then we should aim for specialist classifiers rather than generalists. (2) Our bounds can be used to determine how many classifiers are at least required to achieve a desired ensemble accuracy. Finally, we improve our bounds by considering the mutual information between the true label and the individual classifier's output.