Question

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Exercise 10: 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 are five 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

let event A is have cancer and event B is tested positive

P(B)=P(A)*P(B|A)+P(A')*P(B|A')

=0.05*0.96+0.95*0.05=0.0955

1)

P(has cancer given tested positive) =P(A|B)

P(A|B)=P(A)*P(B|A)/P(B)=0.5026

2)

let event A is have cancer and event B is both test are positive

P(B)=P(A)*P(B|A)+P(A')*P(B|A')

=0.05*0.95*0.95+0.95*0.05*0.05=0.0485

therefore P(have cancer given both test are positive)

P(A|B)=P(A)*P(B|A)/P(B)=0.9510