Abstract:  In this work，the thermodynamic probability problem of N gas molecules distributed in the left and right halves of a rectangular container is rigorously solved using the exact form of Stirlings formula，which quantitatively verifies Boltzmanns statistical interpretation of the microscopic nature of the second law of thermodynamics. It is shown that 95% of the total microscopic states are distributed in the ±Ninterval near the equilibrium state，i.e.，the relative fluctuations are 1/Nwith respect to N. Confusion and errors caused by the common Stirling approximation are discussed.

Key words: the second law of thermodynamics, Stirling formula, Boltzmann entropy, thermodynamic probability