# Probability of a system in the canonical ensemble

In the canonical ensemble, we have the state of system $$x_s$$ and the state of the environment $$x_e$$. The probability of the total system is $$P(x_s,x_e)= const.$$ and that is independent of the states $$x_s$$ and $$x_e$$.

Now the probability to find the system in a certain state $$x_s$$ is $$P(x_s)= \sum_e P(x_s,x_e)$$ which is equal to some constant times the number of accessible states of the environment.

I don't understand how the probability the of system is equal to the sum of states of the environment on the total probability.

The probability $$\rho$$ to find your system $$S$$ with energy $$E_S$$ in a certain state in phase-space is proportional to the phase space volume of the reservoir ("environment") at energy $$E_R$$
$$\rho \propto \Gamma(E_R=E-E_S)$$
with $$E=E_R+E_S$$
Try to look at it this way: For every single state of your system $$S$$ with energy $$E_S$$, you have all states of the reservoir at energy $$E_R=E-E_S$$ available. The sum of all states in phase space is nothing but the phase space volume $$\Gamma$$.