Based on the classical interpretation of probability, the probability for a single particle to be in the $i$th energy state, in an $N$ particle system, should be given by the number of particles in that state, divided by the total number of particles :
Here $n_i$ represents the actual number of particles in the system. However, due to random fluctuations and collisons, the actual number of particles in a particular energy level is never constant and keeps on changing if the total number of particles is finite. By that logic, the 'true' probability of finding a particle at a certain energy level, should be impossible to determine, since we can't sensibly talk about the number of particles in a state, for a finite case anyway.
Hence we use the gibbs/boltzmann distribution, and claim, that the probability of a particle to be in a certain state is given by the following :
However, this is not the exact true probability, is it ? Isn't it technically more like, our best guess of what the true probability of a particle being in state $E_i$ should be ? Since the number of particles in each state keeps on changing, it becomes non-sensical to talk about this 'true' probability of finding the particle in a certain energy state at a time.
So, would it be correct in assuming that the gibbs probability is the average probability or the 'expected' probability of finding a particle in a particular state. Since this is the average probability and not the true one, it becomes impossible to find out the number of particles in that state. Because of this, the number of particles in a state becomes a random variable with a distribution, whose mean is given by $Np(i)$.
So, can we say that in the truest sense of classical probability, the boltzmann probability is average probability of finding a particle in a certain state in the system, only because the true probability keeps on changing as the system undergoes collisions and what not, and the occupancy of a state is never constant ?
In the infinite particle limit, the fluctuations die out, and the actual number of particles become close to the expected number of particles, and so one can claim that the Gibbs probability is approximately equal to the classical probability.
If we could theoretically know the 'true' probability of finding a particle at a certain state, which is impossible ofcourse, we could find the exact number of particles in that state. In that case, the number of particles in the state wouldn't be a distribution, more like a yes or no question, just like picking up coloured balls from a bag - if you know the probability of picking a blue ball from a bag of $100$ balls, you could easily find the number of blue balls. It would be exactly probability multiplied by total number of balls and not some distribution of various possibilities.
But in this case, since you don't know the exact probability, and as the number of particles in the state keep on changing, you can only talk about the expectated number of particles in a state i.e you get a distribution of the total number of particles in a state.
I'm sorry if I'm spending too much time forcing an interpretation of a rather simple problem, but can anyone tell me if my interpretation of the situation correct or not ?