Medical Test - Bayes' Theorem

Problem

A medical test is used to detect a rare disease. Given:

• Prevalence p = 0.01: the probability that a randomly selected person has the disease.
• Sensitivity s = 0.99: the probability the test is positive given the person has the disease (true positive rate).
• Specificity c = 0.95: the probability the test is negative given the person does NOT have the disease (true negative rate).

A person tests positive. What is the probability they actually have the disease?

This classic application of Bayes' theorem frequently appears in quant interviews to test understanding of conditional probability and the base rate fallacy.

Enter your answer as a decimal rounded to 4 decimal places.