1. Massachusetts HIV-Testing Example Test Characteristics ELISA: wrong on 10% of infected samples wrong on 5% of uninfected samples Western Blot: wrong on 5% of infected samples wrong on 5% of uninfected samples Prevalence of virus: 0.2% ELISA Breakdown: Before ELISA neg ELISA pos infected 0.002 0.0002 0.0018 uninfected 0.998 0.9481 0.0499

2. Breakdown after both tests Before ELISA neg WB neg WB pos infected 0.002 0.0002 0.00009 0.00171 uninfected 0.998 0.9481 0.047405 0.002495 Out of 100,000 tested: 171 infected and test positive twice 249.5 uninfected and test positive twice 180 infected and test positive on ELISA 4990 uninfected and test positive on ELISA

3. Infected Uninfected.002.998

4. Infected Uninfected.002.998 90% 10% Infected & ELISA Pos Infected & ELISA Neg

5. Infected Uninfected.002.998 90% 10% 5% 95% Infected & ELISA Pos Infected & ELISA Neg Uninfected & ELISA Pos Uninfected & ELISA Neg

6. Infected Uninfected.002.998 90% 10% 5% 95% Infected & ELISA Pos Infected & ELISA Neg Uninfected & ELISA Pos Uninfected & ELISA Neg 95% 5% Infected & WB Pos Infected & WB Neg

7. Infected Uninfected.002.998 90% 10% 5% 95% Infected & ELISA Pos Infected & ELISA Neg Uninfected & ELISA Pos Uninfected & ELISA Neg 95% 5% 95% Infected & WB Pos Infected & WB Neg Uninfected & WB Pos Uninfected & WB Neg

8. Concept AIDS Example Die Example sample space “low-risk” population {1,2,3,4,5,6} event A = {infected individuals} A={2,4,6} B = {positive on ELISA} B={1,2} probability P(A) = 0.002 P(A) = 1/2 P(B) = 0.0517 P(B) = 1/3 complement not A = {not infected} not A = {1,3,5} intersection (A and B) = (A and B) = {infected and ELISA pos} {2} union (A or B) = (A or B) = {infected or ELISA pos} {1,2,4,6} mutually exclusive A1 = {test pos on both} A1 = {even} A2 = {test neg on both} A2 = {odd}

9. complement: P(not A) = 1 - P(A) Addition rule: P(A or B) = P(A) + P(B) - P(A and B) Conditional probability: P(A|B) = P(A and B)/P(B) Multiplication rule: P(A and B) = P(A)P(B|A)