In a certain community, eight percent of all adults over age 50 have diabetes. If a health service in this community correctly diagnosis 95% of all persons with diabetes as having the disease and incorrectly diagnoses ten percent of all persons without diabetes as having the disease, find the probabilities that:

Complete question is;

In a certain community, 8% of all people above 50 years of age have diabetes. A health service in this community correctly diagnoses 95% of all person with diabetes as having the disease, and incorrectly diagnoses 10% of all person without diabetes as having the disease. Find the probability that a person randomly selected from among all people of age above 50 and diagnosed by the health service as having diabetes actually has the disease.

Answer:

P (has diabetes | positive) = 0.442

Step-by-step explanation:

Probability of having diabetes and being positive is;

P (positive & has diabetes) = P (has diabetes) * P (positive | has diabetes)

We are told 8% or 0.08 have diabetes and there's a correct diagnosis of 95% of all the persons with diabetes having the disease.

Thus;

P (positive & has diabetes) = 0.08 * 0.95 = 0.076

P (negative & has diabetes) = P (has diabetes) * (1 - P (positive | has diabetes)) = 0.08 * (1 - 0.95)

P (negative & has diabetes) = 0.004

P (positive & no diabetes) = P (no diabetes) * P (positive | no diabetes)

We are told that there is an incorrect diagnoses of 10% of all persons without diabetes as having the disease

Thus;

P (positive & no diabetes) = 0.92 * 0.1 = 0.092

P (negative &no diabetes) = P (no diabetes) * (1 - P (positive | no diabetes)) = 0.92 * (1 - 0.1)

P (negative &no diabetes) = 0.828

Probability that a person selected having diabetes actually has the disease is;

P (has diabetes | positive) = P (positive & has diabetes) / P (positive)

P (positive) = 0.08 + P (positive & no diabetes)

P (positive) = 0.08 + 0.092 = 0.172

P (has diabetes | positive) = 0.076/0.172 = 0.442