How do calculate the prob of being in a state in therm eq?

[eprparadox]

Homework Statement

A pore has three configurations with the energy levels shown. The pore is in thermal equilibrium with the surroundings at temperature T . Find the probabilities $p_1$, $p_2$, $p_3$ of being in each level. Each level has only one microstate associated with it.

Also, state 1 corresponds to energy of 0, state 2 to energy of $U_0$ and state 3 to energy of $2U_0$.

Homework Equations

The Boltzmann factor = $e^{-(U_s - U_r)/kT}$

The probability of being in a state is equal to $$p_i = \frac{e^{\frac{-E_i}{kT}}}{\sum{\frac{-E_i}{kT}}}$$

where the denominator is the sum over all states.

The Attempt at a Solution

This is for self study and not for a course and so I'm really trying to explain these solutions to myself.

In terms of an attempt, I know that

$$\frac{p_2}{p_1} = \frac{\Omega(2)}{\Omega(1)}$$

But $$\Omega(2) = \Omega(1) = 1$$ so the probability of being in any state is the same. And since the we have 3 energy states, I get that $$p_1 = p_2 = p_3 = \frac{1}{3}$$.

Is that correct?

And if so, how can I use the Boltzmann factor to arrive at the same answer.

 

[haruspex]

The principle of equally likely microstates only applies where those microstates are sufficiently alike. In particular, they must have the same energy. That is not the case here.
Use the relevant equations you quote (Boltzmann) to find the probabilities from the energies.