A vessel of volume \(V^{(0)}\) contains \(N^{(0)}\) molecules. Assuming that there is no correlation whatsoever between the locations of the various molecules, calculate the probability, \(P(N, V)\), that a region of volume \(V\) (located anywhere in the vessel) contains exactly \(N\) molecules.

(a) Show that \(\bar{N}=N^{(0)} p\) and \((\Delta N)_{\text {r.m.s. }}=\left\{N^{(0)} p(1-p)ight\}^{1 / 2}\), where \(p=V / V^{(0)}\).

(b) Show that if both \(N^{(0)} p\) and \(N^{(0)}(1-p)\) are large numbers, the function \(P(N, V)\) assumes a Gaussian form.

(c) Further, if \(p \ll 1\) and \(N \ll N^{(0)}\), show that the function \(P(N, V)\) assumes the form of a Poisson distribution:

\[
P(N)=e^{-\bar{N}} \frac{(\bar{N})^{N}}{N !}
\]