A new test has been developed to detect a particular type of cancer. A medical r

Question

A new test has been developed to detect a particular type of cancer. A medical researcher selects a random sample of 1,000 adults and finds (by other means) that 4% have this type of cancer. Each of the 1,000 adults is given the new test and it is found that the test indicates cancer in 99% of those who have it and in 1% of those who do not. Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer? What is the probability of a person having cancer given that the test does not indicate cancer.

Please explain the answer was calculated.

Explanation / Answer

Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?

Let

D = has cancer disease
P = tests positive

As by Bayes' Rule,

P(P) = P(D) P(P|D) + P(D') P(P|D') = 0.04*0.99 + (1-0.04)*0.01 = 0.0492

Hence,

P(D|P) = P(D) P(P|D)/P(P) = 0.04*0.99/0.0492 = 0.804878049 [ANSWER]
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b) What is the probability of a person having cancer given that the test does not indicate cancer?

P(D|P') = P(D) P(P'|D)/P(P') = 0.04*(1-0.99)/(1-0.0492) = 0.000420698 [ANSWER]

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