Problem Statement
A diagnostic test is $99\%$ effective in detecting a disease. It gives $0.5\%$ false positive. If $0.1\%$ of the population has the disease, what is the probability a person has the disease given a positive test?
Verified Solution & Marking Scheme
Define Priors
$P(E) = 0.001, \, P(\bar{E}) = 0.999$.
Likelihoods
$P(A|E) = 0.99, \, P(A|\bar{E}) = 0.005$.
Bayes' Theorem
$P(E|A) = \frac{0.001 \times 0.99}{(0.001 \times 0.99) + (0.999 \times 0.005)} = \frac{0.00099}{0.005985} = \frac{22}{133} \approx 0.165$