Question

To prevent developing a certain type of cancer, Health Canada organize a di-agnostic test. The false positive and negative rates for this diagnostic test arefive percent and four percent, respectively. Assume that five percent of the population has this type of cancer.1. If a randomly selected person has a positive test result, what is the prob-ability that the person has this type of cancer?2. The test is administrated twice on the same patient. Assume that the test results are independent from each other, conditionally on the patient’s disease status. Given that both tests results are positive, what is the probability that the patient has this type of cancer?

Answer 1

1)

P(tested positive)=P(have cancer )*P(tested positive |have cancer)+P(not have cancer )*P(tested positive |not have cancer)=0.05*(1-0.04)+(1-0.05)*0.05 =0.0955

therefore P(have cancer |tested positive)

=P(have cancer )*P(tested positive |have cancer)/P(tested positive)=0.05*(1-0.04)/0.0955 =0.5026

2)

P(both test are positive)

=P(have cancer )*P(twice tested positive |have cancer)+P(not have cancer )*P(twice tested positive |not have cancer)

=0.05*(1-0.04)*(1-0.04)+(1-0.05)*0.05*0.05 =0.0485

P(have cancer |both test are positive) =0.05*(1-0.04)*(1-0.04)/0.0485=0.9510