In the derivation of the Boltzmann distribution they consider a system $A$, enclosed by a diathermal wall in a heat reservoir $R$. Then they calculate the probability that the system $A$ is in an energy state $E_r$, given that the reservoir has a temperature $T$ and energy $E_0$ and the two systems are in thermal equilibrium.
I don't understand though why we can speak about probabilities. I would rather expect that we can calculate the energy of $A$ exactly if indeed the systems are in thermal equilibrium. Because the temperature of $R$ is then exactly $T$, so we can calculate its energy exactly (this last step is what I think we can do, though I'm not completely sure). So I wonder: is the derivation indeed for thermal equilibrium, like they state explicitly in my book. Or is that just a confusing mistake, and does the Boltzmann distribution give the probability if the system $A$ is in contact with $R$ but not necessarily in equilibrium with it? In the last case the derivation seems actually to make sense, but I'm worrying about the fact that they explicitly say the it is for thermal equilibrium. The fluctuations in energy cause then also fluctuations in temperature, so according to this, there would also never be exactly thermal equilibrium (so I'm not exactly sure the last case should be the right one either).