6. In a certain community, 8 percent of all adults over 50 years old have diabet

Question

6. In a certain community, 8 percent of all adults over 50 years old have diabetes. If a health service in this community correctly diagnoses 95 percent of all persons with diabetes as having the disease and incorrectly diagnoses 2 percent of all persons without diabetes as having the disease, find the probabilities that (a) the community health service will diagnose an adult over 50 as having diabetes; (b) a person over 50 diagnosed by the health service as having diabetes actually has the disease.

Explanation / Answer

Question 6

To make the question easy, i will define two events first

A = Adults over 50 year old have diabetes

B = Adults over 50 year old founded positive to diabetes

so as by given information

Pr(Adults having diabetes) = P(A) = 0.08

Pr(Positive result when a person is diabetic) = Pr(B l A) = 0.95

Pr(incorrectly diagnoses a person diabetic even he is not diabetes) = Pr(B l Ac) = 0.02

(a) Pr(A person will diagnose) = Pr(B) = Pr(B l A) * Pr(A) + Pr(B l Ac) * Pr(Ac) = 0.95 * 0.08 + 0.02 * 0.92 = 0.0944

(b) Pr(Actually have the disease l diagonsed as diabetic) = Pr(A l B) = Pr(B l A) * Pr(A) / [Pr(B l A) * Pr(A) + Pr(B l Ac) * Pr(Ac) ] = 0.95 * 0.08/ 0.0944 = 0.8051

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