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We prove that information-theoretic maximum entropy (MaxEnt) approach tocanonical ensemble is mathematically equivalent to the classic approach ofBoltzmann, Gibbs and Darwin-Fowler. The two approaches, however, "interpret" asame mathematical theorem differently; most notably observing mean-energy inthe former and energy conservation in the latter. However, applying the sameMaxEnt method to grand canonical ensemble fails; while carefully following theclassic approach based on Boltzmann's microcanonical {\em equal probability apriori} produces the correct statistics: One does not need to invoke quantummechanics; and there is no Gibbs paradox. MaxEnt and related minimum relativeentropy principle are based on the mathematical theorem concerning largedeviations of rare fluctuations. As a scientific method, it requires classicmechanics, or some other assumptions, to provide meaningful {\em priordistributions} for the expected-value based statistical inference. A naiveassumption of uniform prior is not valid in statistical mechanics.

Maximum Entropy Principle, Equal Probability a Priori and Gibbs Paradox