A rare disease affects 1/1000 individuals. This disease is difficult to diagnose

Question

A rare disease affects 1/1000 individuals. This disease is difficult to diagnose, but one way is to test for an elevated level of a particular enzyme. Levels of this enzyme vary within the population, so the test is not always exact. The following table gives the probability of different test results given the patients disease status.

Patient has disease

Patient does not have disease

Positive test (elevated)

80%

10%

Negative test (normal)

20%

90%

a.     What is the false negative rate of this test?

b.     What is the probability of a patient receiving a positive test?

c.     The test gave a positive result. What is the probability that the patient has the disease?  

Patient has disease

Patient does not have disease

Positive test (elevated)

80%

10%

Negative test (normal)

20%

90%

Explanation / Answer

a)false negative rate of this test =P(negaitive test given patient has disease) =20%  

b)probability of a patient receiving a positive test =P|(have disease and test positive+not have disease and test positive)

=(1/1000)*0.80+(1-1/1000)*0.1 =0.1007

c)probability that the patient has the disease given test posiitve

=P(have disease and test positive)/P(test positive)=(1/1000)*0.80/0.1007=0.007944

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