1) Explain each step Disease A occurs in 0.1% of the population. If a person doe

Question

1) Explain each step

Disease A occurs in 0.1% of the population. If a person does NOT have the disease and takes a test for the disease, the test correctly indicates that they don’t have the disease 96% of the time. If a person has the disease and takes the same test, the test correctly indicates that they do have the disease 98% of the time. Given that you’ve taken the test and have tested positive for disease A, what is the probability you actually have disease A? Given that you’ve taken the test and have tested negative for disease A, what is the chance that you have the disease anyway?

Explanation / Answer

Let B1 be the event that a person has the disease, let B2 be the event that a person does not have the disease and let A be the event that the test result is positive. Then

P(B1) = 0.001, P(B2) = 1 P(B1) = 0.999, P(A|B1) = 0.98, P(A|B2) = 1-0.96= 0.04

(a) P(B1|A) = P(A|B1)P(B1) /(P(A|B1)P(B1) + P(A|B2)P(B2)) = (0.98)(0.001)/ ((0.98)(0.001) + (0.04)(.999) = 0.02394

(b) Now let A be the event that the test result is negative here, P(A|B1) = 1- 0.98 = 0.02 and P(A|B2) = 0.96

P(B1|A) = P(A|B1)P(B1) /(P(A|B1)P(B1) + P(A|B2)P(B2)) = (0.02)*(0.001)/((0.02)(0.001)+(0.999)(0.96)) = 0.00002085

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