Using the same method as we did with Boltzmann Factors, we consider a system in thermal and diffusive equilibrium with another system.

P(N_{1},ε_{1})/P(N_{2},ε_{2}) = g(N_{0} - N_{1}, U_{0} - ε_{1})/g(N_{0} - N_{2}, U_{0} - ε_{2})

And by the exact same method as our derivation of Boltzmann factors (no surprises here):

P(N_{1},ε_{1})/P(N_{2},ε_{2}) = exp((N_{1}μ - ε_{1})/T)/exp((N_{2}μ - ε_{2})/T)

Which we will use in the exact same way to create the Grand Partition Function.