Conditional probability and Bayes' theorem. A medical screening test for a rare disease has a sensitivity (true positive rate) of 98 % and a specificity (true negative rate) of 95 %. The disease prevalence in the population is 2 %. (a) Construct a probability tree showing all possible outcomes (disease/healthy and test positive/test negative) and the probability of each branch. (b) Using Bayes' theorem, compute the posterior probability that a randomly selected individual who tests positive actually has the disease. Interpret the result. (c) If the test is repeated independently for an individual who tests positive the first time, determine the probability that they test positive on both tests and actually have the disease. (d) Discuss how lowering the disease prevalence (e.g., 0.5 %) affects the positive predictive value. Should screening be recommended for very rare diseases