 Solution

Let $Z=1$ mean membership in the structural-zero component. Under the fitted zero-inflated Poisson regression, $P(Y=0\mid Z=1)=1$, whereas $P(Y=0\mid Z=0)=e^{-\mu }$. By Bayes' theorem, the posterior structural-zero probability isUsing the printed predictions for this patient givesThe fitted conditional probability is about $50.0\%$. It is larger than the prior fitted structural-zero probability $0.4448758$, because observing no episodes increases the probability of latent membership in that component. The interpretation “never at risk” is the model's structural class; an observed six-month zero alone does not identify the class with certainty.