A diagnostic test has 95% sensitivity and 90% specificity. Disease prevalence is 2%. What is the probability that a person who tests positive actually has the disease? A) about 16% B) about 66% C) about 90% D) about 95%

The probability that a person who tests positive actually has the disease is approximately 16%. Here's how to calculate it: 1. Define the given probabilities: Sensitivity (P(Test Positive | Disease)) = 0.95 Specificity (P(Test Negative | No Disease)) = 0.90 Prevalence (P(Disease)) = 0.02 2. Calculate related probabilities: P(No Disease) = 1 - P(Disease) = 1 - 0.02 = 0.98 P(Test Positive | No Disease) = 1 - Specificity = 1 - 0.90 = 0.10 (This is the false positive rate) 3. Calculate the overall probability of testing positive (P(Test Positive)) using the law of total probability: P(Test Positive) = P(Test Positive | Disease) P(Disease) + P(Test Positive | No Disease) P(No Disease) P(Test Positive) = (0.95 0.02) + (0.10 0.98) P(Test Positive) = 0.019 + 0.098 P(Test Positive) = 0.117 4. Calculate the Positive Predictive Value (PPV), which is P(Disease | Test Positive) using Bayes' Theorem: P(Disease | Test Positive) = [P(Test Positive | Disease) * P(Disease)] / P(Test Positive) P(Disease | Test Positive) = 0.019 / 0.117 P(Disease | Test Positive) ≈ 0.1624 5. Convert to percentage: 0.1624 * 100% = 16.24% This value is closest to A) about 16%.