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The AdS/CFT correspondence is useful primarily when the number of colours, Nc, characterising the boundary field theory, is “large”, and when the mass of the bulk black hole that is usually present is “large” relative to the bulk Planck mass. But this prompts two questions: first, can these large numbers be estimated, even very approximately, in a given application? Second: if these quantities are themselves computed holographically from physical data constraining the field theory, is this computation self-consistent, in the sense that it actually produces large numbers — an outcome which is far from obvious? Here we consider these questions in the case of the application of holographic techniques to the study of the quark-gluon plasma. We find that holography in this case is able to generate estimates of the dimensionless numbers in question, and, very remarkably, they are indeed large, despite the fact that the dimensionless input data are of order unity.