(a) When a system of \(N\) oscillators with total energy \(E\) is in thermal equilibrium, what is the probability \(p_{n}\) that a particular oscillator among them is in the quantum state \(n\) ? [Hint: Use expression (3.8.25).]

Show that, for \(N \gg 1\) and \(R \gg n, p_{n} \approx(\bar{n})^{n} /(\bar{n}+1)^{n+1}\), where \(\bar{n}=R / N\).

(b) When an ideal gas of \(N\) monatomic molecules with total energy \(E\) is in thermal equilibrium, show that the probability of a particular molecule having an energy in the neighborhood of \(\varepsilon\) is proportional to \(\exp (-\beta \varepsilon)\), where \(\beta=3 N / 2 E\).