Degeneracy "increases the probability of higher energy state population" A week ago, my lecturer was talking about how degeneracy can be implemented into the Boltzmann Equation:
$$\frac{N_j}{N_i} = \frac{g_j}{g_i}e^{\frac{- \triangle E}{kT}}$$
and he said that for any number $J$, the number of degenerate energy levels was calculated to be $2J+1$. 
This all made sense. 
However, in the problem class, he said that degeneracy increases the probability of higher energy states being populated as the electrons have more of a chance of reaching the higher states. 
This didn't make sense to me. I understood that the number of degeneracy levels increased as it was dependent on $J$ but his explanation didn't make sense of all, and Google was of little help as it didn't mention anything about the increased probability at all. 
Can someone explain what he meant?
 A: The probability of each state is proportional to the Boltzmann factor $\exp(-E/kT)$. Now suppose that there are $g$ states, all having the same energy. The total probability of having an energy $E$ is the sum of the individual state probabilities, giving a result proportional to $g\exp(-E/kT)$. It's just a way of grouping states together into levels which have the same energy.
The formula you were given for $g$ applies in some specific cases, for instance when the states are defined by two quantum numbers $J$ and $M$, when there are $2J+1$ possible values of $M$ for each $J$, such as $M=J, J-1, J-2, \ldots -J$, and when the energy does not depend on $M$, just on $J$. In different cases, the degeneracy might depend in different ways on energy, but it is quite common for it to increase with increasing energy. For this reason, the probability of finding the system with higher energies gets weighted more heavily at higher energies: there are simply more states at each energy, as the energy goes up.
Sometimes it is convenient to express properties of the system as a formula involving a sum over energies, rather than a sum over states. In such a formula, the degeneracy will appear as well as the Boltzmann factor, for each energy level. 
I'll just mention in passing that for systems having a continuous distribution of energies, the sum becomes an integral over energy, and the degeneracy becomes a density of states $g(E)$, i.e. a number of states per unit energy. The basic idea is the same as for discrete energies.
