1. **Problem statement:** We have a population where 8% actually have COVID-19. The test has a true positive rate of 92% and a false positive rate of 6%. We want to find: 1. Probability a person tests positive. 2. Probability a person tests negative. 3. Probability the test diagnosis is correct. 2. **Step 1: Define probabilities.** - Let $P(C)$ = probability person has COVID = 0.08 - Let $P(H)$ = probability person is healthy = 1 - 0.08 = 0.92 - True positive rate $P(T+|C)$ = 0.92 - False positive rate $P(T+|H)$ = 0.06 3. **Step 2: Calculate the probability that a person tests positive.** The total probability of testing positive $P(T+)$ is calculated by the law of total probability: $$ P(T+) = P(T+|C)P(C) + P(T+|H)P(H) $$ Substitute values: $$ P(T+) = (0.92)(0.08) + (0.06)(0.92) = 0.0736 + 0.0552 = 0.1288 $$ 4. **Step 3: Calculate the probability that a person tests negative $P(T-)$** Since testing negative is the complement of testing positive: $$ P(T-) = 1 - P(T+) = 1 - 0.1288 = 0.8712 $$ 5. **Step 4: Calculate the probability that the doctor's diagnosis is correct.** This means either correctly identifying sick people (true positives) or correctly identifying healthy people (true negatives). True negative rate $P(T-|H) = 1 - P(T+|H) = 1 - 0.06 = 0.94$ So, overall accuracy: $$ P(correct) = P(T+ \cap C) + P(T- \cap H) = P(T+|C)P(C) + P(T-|H)P(H) $$ Substitute: $$ P(correct) = (0.92)(0.08) + (0.94)(0.92) = 0.0736 + 0.8648 = 0.9384 $$ 6. **Final answers:** - Probability testing positive: $0.1288$ - Probability testing negative: $0.8712$ - Probability diagnosis is correct: $0.9384$