Question

Ten percent of a population suffers from a serious disease. A person suspected of the disease is given two independent tests. Each test makes a correct diagnosis 90% of the  time. Find the probability that the person really has the illness a. given that both tests are positive or b. given that only one test is positive.
Please use calculus where possible and preferably not Excel.

Answer 1

a)P(both test are posiitve)=P(has illness and both test are positive)+P(not have illness and both test are positive)=0.1*0.9*0.9+(1-0.1)*(1-0.9)*(1-0.9)=0.09

hence P(has illness given both test are positive)=P(has illness and both test are positive)/P(both test are positive)=0.1*0.9*0.9/0.09=0.9

b)

P(one test positive)=P(has illness and one test is positive)+P(not have illness and one test is positive)

=0.1*2*0.9*0.1+(1-0.1)*2*0.9*0.1

=0.18

hence P(has illness given one test is positive)=P(has illness and one test is positive)/P(one test positive)=0.1*2*0.9*0.1/0.18 =0.1