D. Wo heads and a tail in that order? 13, In a certain community, 15% of all adu

Question

D. Wo heads and a tail in that order? 13, In a certain community, 15% of all adults over 50 have diabetes. If a health service in this community correctly diagnoses 95% of all persons with diabetes as having the disease and incorrectly diagnoses 10% 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

Here we are given that 15% of all adults have diabetes. Therefore P( diabetes ) = 0.15

Also we are given that for the test,

P( positive | diabetes ) = 0.95 and P( positive | no diabetes ) = 0.1

a) Probability that the diagnosis be positive is computed using the law of total probability as:

P( positive ) = P( positive | diabetes )P( diabetes ) + P( positive | no diabetes )P( no diabetes )

P( positive ) = 0.95*0.15 + 0.1*(1 - 0.15) = 0.2275

Therefore 0.2275 is the required probability here that the person would be diagnosed positive

b) Probability that a person diagnosed positive actually have the disease is computed using the Bayes theorem as:

P( diabetes | positive ) = P( positive | diabetes )P( diabetes ) / P( positive) = 0.95*0.15 / 0.2275 = 0.6264

Therefore 0.6264 is the required probability here.

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