Question

The sensitivity of a medical test refers to the test’s ability to correctly detect ill patients who have the condition. Mathematically, sensitivity is equivalent to the probability that the test indicates a patient is ill given that the patient is ill. The specificity of a medical test refers to the test’s ability to correctly detect that a healthy patient does not have the condition. Mathematically, specificity is equivalent to the probability that the test indicates the patient is healthy given that the patient is healthy.

If the number of cases of the flu for men aged 35 to 40 is 128.4 per 100,000 men. For the purposes of this problem, the sensitivity of the test is 0.97 and the specificity of the test is 0.645. Let BC+ indicate a male has the flu and BC- indicate a male does not have the flu. Let M+ indicate an abnormal or suspicious test and M- indicate a test that shows no sign of the flu. Calculate the probability of having the flu given a abnormal test, P(BC+|M+) for an average male who is 37 years old.

Answer 1

From given information we have

P(BC+) = 128.4 / 100000 = 0.001284

By the complement rule,

P(BC-) = 1 - P(BC+) = 0.998716

And we have

P(M+ | BC+) = 0.97

P(M- | BC-) = 0.645

By the complement rule,

P(M+ | BC-) = 1 - P(M- | BC-) = 1 - 0.645 = 0.355

Using Baye's theorem , the probability of having the flu given a abnormal test, P(BC+|M+) for an average male who is 37 years old is

P(BC+ | M+) = [ P(M+ | BC+) * P(BC+) ] / [ P(M+ | BC+) * P(BC+) + P(M+ | BC-) * P(BC-) ] = [ 0.97 * 0.001284 ] / [ 0.97 * 0.001284 + 0.355 * 0.998716 ] = 0.00124548 / 0.35578966 = 0.00350061

Answer: 0.00350061