This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the explanation and solution for the diagnostic test problem using Bayes' theorem: Step 1: Identify the given probabilities. Let D be the event that a person has the disease, and + be the event that the test is positive. Sensitivity (probability of testing positive given the disease): P(+|D) = 0.95 False positive rate (probability of testing positive given no disease): P(+| D) = 0.10 Disease prevalence (prior probability of having the disease): P(D) = 0.20 Step 2: Calculate the probability of not having the disease. The probability of not having the disease, P( D), is 1 - P(D). P( D) = 1 - P(D) = 1 - 0.20 = 0.80 Step 3: Apply Bayes' theorem. We want to find the probability of having the disease given a positive test, P(D|+). Bayes' theorem states: P(D|+) = (P(+|D) · P(D))/(P(+|D) · P(D) + P(+| D) · P( D)) Step 4: Substitute the values and calculate. Substitute the known probabilities into the formula: P(D|+) = (0.95 × 0.20)/((0.95 × 0.20) + (0.10 × 0.80)) Calculate the terms: P(D|+) = (0.19)/(0.19 + 0.08) P(D|+) = (0.19)/(0.27) P(D|+) ≈ 0.7037 Step 5: Round to the nearest given option. Rounding 0.7037 to two decimal places gives 0.70. The probability of having the disease given a positive test is approximately 0.70. The correct option is d) 0.70. That's 2 down. 3 left today — send the next one.