6. A certain disease affects about 1 out of 1000 people. There is a test to chec

Question

6. A certain disease affects about 1 out of 1000 people. There is a test to check whether the person has the disease. The test is quite accurate. In particular, we know that the probability that the test result is positive (suggesting the person has the disease), given that the person does not have the disease, is only 2 percent; the probability that the test result is negative (suggesting the person does not have the disease), given that the person has the disease, is only 1 percent. A random person gets tested for the disease and the result comes back positive. What is the probability that the person has the disease?

Explanation / Answer

let probability of disease =P(D)=1/1000 =0.001

and probability of test postiive =P(T)

probability that test is positive given not have the disease =P(T|Dc) =0.02

probability of test negative given has the disease =P(Tc|D) =0.01

hence probability of test positive given has the disease =P(T|D) =1-P(Tc|D) =1-0.01 =0.99

hence probability of test result positive =P(T)=P(D)*P(T|D)+P(Dc)*P(T|Dc) =0.001*0.99+(1-0.001)*0.02=0.02097

hence probability of person having the disease given test result positive =P(D|T) =P(D)*P(T|D)/P(T)

=0.001*0.99/0.02097 =0.0472

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