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%

• Sensitivity (P(Test Positive | Disease)) = 0.95
• Specificity (P(Test Negative | No Disease)) = 0.90
• Prevalence (P(Disease)) = 0.02

• 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)

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

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

0.1624 * 100% = 16.24%

This value is closest to A) about 16%.