The probability that a system in the grand canonical ensemble has exactly \(N\) particles is given by

\[
p(N)=\frac{z^{N} Q_{N}(V, T)}{\mathcal{Q}(z, V, T)}
\]

Verify this statement and show that in the case of a classical, ideal gas the distribution of particles among the members of a grand canonical ensemble is identically a Poisson distribution. Calculate the root-mean-square value of \((\Delta N)\) for this system both from the general formula (4.5.3) and from the Poisson distribution, and show that the two results are the same.