The random energy model, originally used to analyze the physics of spin glasses, has been employed to explore what makes a protein a good folder versus a bad folder. In earlier work, the ratio of the folding temperature over the glass-transition temperature was related to a statistical measure of protein energy landscapes denoted as the foldability F. It was posited and subsequently established by simulation that good folders had larger foldabilities, on average, than bad folders. An alternative hypothesis, equally verified by protein folding simulations, was that it is the energy gap Delta between the native state and the next highest energy that distinguishes good folders from bad folders. This duality of measures has led to some controversy and confusion with little done to reconcile the two. In this paper, we revisit the random energy model to derive the statistical distributions of the various energy gaps and foldability. The resulting joint distribution allows us to explicitly demonstrate the positive correlation between foldability and energy gap. In addition, we compare the results of this analytical theory with a variety of lattice models. Our simulations indicate that both the individual distributions and the joint distribution of foldability and energy gap agree qualitatively well with the random energy model. It is argued that the universal distribution of and the positive correlation between foldability and energy gap, both in lattice proteins and the random energy model, is simply a stochastic consequence of the "thermodynamic hypothesis."