CMA Foundation · Fundamentals of Business Mathematics and Statistics · Probability
In a town, 2% of people have a certain disease. A screening test gives a positive result for 90% of people who have the disease and also for 10% of people who do not have it. A person tests positive. What is the probability that the person actually has the disease?
The probability is 9/58, about 15.5%. Of the population, 0.018 are diseased and test positive, while 0.098 are healthy and test positive. The total positive proportion is 0.116, so the diseased share is 0.018/0.116 = 9/58. The low prevalence causes many false positives.
- A9/58Correct
- B9/50
- C9/10
- D1/10
Explanation
P(disease and positive) = 0.02 x 0.90 = 0.018. P(no disease and positive) = 0.98 x 0.10 = 0.098. Total positive = 0.116. Posterior = 0.018/0.116 = 9/58. The option 9/50 comes from wrongly using 0.10 as the denominator, and 9/10 is only the test's sensitivity.
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