1. **Problem statement:** A patient undergoes two independent diagnostic tests for the same disease. Each test has 90% sensitivity and 95% specificity. We want to find: a) The probability that both tests return positive for a person who has the disease. b) The probability that both tests return positive for a person who does not have the disease. 2. **Definitions and formulas:** - Sensitivity is the probability that the test is positive given the person has the disease: $P(\text{Test}^+|\text{Disease}) = 0.9$. - Specificity is the probability that the test is negative given the person does not have the disease: $P(\text{Test}^-|\text{No Disease}) = 0.95$. Since the tests are independent, the joint probability of both tests being positive is the product of their individual probabilities. 3. **Calculations:** **a) Both tests positive given disease:** $$ P(\text{Both}^+|\text{Disease}) = P(\text{Test}_1^+|\text{Disease}) \times P(\text{Test}_2^+|\text{Disease}) = 0.9 \times 0.9 = 0.81 $$ **b) Both tests positive given no disease:** First, find the probability that a test is positive given no disease: $$ P(\text{Test}^+|\text{No Disease}) = 1 - P(\text{Test}^-|\text{No Disease}) = 1 - 0.95 = 0.05 $$ Then, $$ P(\text{Both}^+|\text{No Disease}) = 0.05 \times 0.05 = 0.0025 $$ 4. **Interpretation:** - For a person with the disease, there is an 81% chance both tests will be positive. - For a person without the disease, there is a 0.25% chance both tests will be falsely positive. **Final answers:** - a) $0.81$ - b) $0.0025$