Consider an $\mathrm{M} / \mathrm{M} / 1 / \mathrm{K}$ queueing system. Let $q_n$ be the probability there are $n$ customers in the system just before a customer arrival that actually enters the system; that is, $q_n$ is the probability that there an $n$ customers in the system when an arrival is about to occur. Thus, $q_n=P\left[A_n \mid A\right]$ for $n=0,1,2, \ldots, K-1$, where $A_n$ is the event that there are $n$ customers in the system and $A$ is the event that an arrival is about to occur. Use Bayes' theorem (Theorem 2.4.3) to prove that
$$
q_n=\frac{p_n}{1-p_K}, \quad n=0,1, \ldots, K-1 .
$$