Compute posterior for accurate-but-rare classifier

Within a population, 5% of users are 'bad' and 95% are 'good'. A screening model has sensitivity 0.95 for predicting 'bad' ($P(\text{pred}=\text{bad} \mid \text{bad})=0.95$) and specificity 0.95 ($P(\text{pred}=\text{good} \mid \text{good})=0.95$). (a) When the model predicts 'bad' for a user, compute $P(\text{user is actually bad})$. (b) When the model predicts 'good', compute $P(\text{user is actually good})$. (c) Explain how these posteriors change with the prevalence and why this illustrates the base-rate effect.

Overview: This question tests understanding of Bayes' theorem and probabilistic reasoning for interpreting classifier outputs, specifically computing posterior probabilities given sensitivity, specificity, and prevalence.