This might be a silly question, but I don't see the equivalence relation between these two equations. Could somebody explain to me how to derive one from the other? Thanks in advance!
Citing Wikipedia here,
In what has been called the fundamental assumption of statistical thermodynamics or the fundamental postulate in statistical mechanics, the occupation of any microstate is assumed to be equally probable (i.e. Pi = 1/Ω, where Ω is the number of microstates); this assumption is usually justified for an isolated system in equilibrium.
Since $P_i = 1/\Omega$, obviously $\ln P_i =-\ln\Omega$ and since $P_i$ is probability, then the $\Sigma_i P_i=1$