the fundamental postulate of statistical mechanics
This is the single assumption on which the whole edifice rests. For an isolated system in equilibrium, every accessible microstate is equally likely. No microstate is special; given only the fixed energy, volume and particle number, nature shows no preference among the microscopic arrangements consistent with them.
Precisely: for an isolated system with fixed (E, V, N), if there are W accessible microstates, all with energy in a thin shell from E to E + dE, then each occurs with probability 1/W. This equal-weighting defines the microcanonical ensemble. The postulate is motivated by, but not derived from, Liouville's theorem, which says that a uniform probability density on the energy shell stays uniform under Hamiltonian flow, and by the ergodic idea that a long time average equals an average over that uniform ensemble.
From this one postulate flow the Boltzmann entropy S = k_B ln W and, through it, all of thermodynamics. Honesty demands we call it a postulate: it is justified after the fact by its success, not proved from mechanics. Ergodicity, the claim that time averages equal ensemble averages, is a deep and not fully general condition; integrable, glassy, or otherwise non-mixing systems can violate it in practice.
A fair die has six equally likely faces; the postulate says an isolated system's accessible microstates are like the faces of an unimaginably many-sided fair die.
Equal a priori probability: no accessible microstate is favoured over another.
Equal probability holds only among accessible microstates at fixed energy; a system in contact with a heat bath follows the Boltzmann distribution, not uniform weighting.