Bayesian Medical Diagnostic Testing

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Bayes' Theorem.

Your Task
A rare disease has a prevalence of $0.1\%$ in a population ($\mathbb{P}(D) = 0.001$). A diagnostic test has a true positive rate (sensitivity) of $99\%$ ($\mathbb{P}(+\mid D) = 0.99$) and a false positive rate of $2\%$ ($\mathbb{P}(+\mid D^c) = 0.02$). A randomly chosen individual tests positive. Calculate the posterior probability that the individual actually has the disease.
Constraints

• Express answer as an exact fraction and as a floating-point percentage.