A new, simple test has been developed to detect a particular type of cancer. The

Question

A new, simple test has been developed to detect a particular type of cancer. The test must be evaluated before it is put into use. A medical researcher selects a random sample of 2,000
adults and finds (by other means) that 2 % have this type of cancer. Each of the 2,000 adults is given the test, and it is found that the test indicates cancer in 98 %of those who have it and in 2 %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? Of a person having cancer given that the test does not indicate cancer?

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

(Use Bayes' Formula and a diagram to calculate the probability of a person having cancer given that the test indicates cancer. Form a tree diagram.)

?(Round to three decimal places as? needed.)

B:What is the probability of a person having cancer given that the test does not indicate cancer?

(This result is given by P(C|T'). By Bayes' formula, this equals the product of branch probabilities leading to T' through C divided by the sum of all branch products leading to T'.)

?(Round to three decimal places as? needed.)

Explanation / Answer

P(test indicate cancer)=P(have cancer and test indicate cancer+does not have cancer and test indicate cancer)

=0.02*0.98+0.98*0.02=0.0392

therfore probability of a randomly chosen person having cancer given that the test indicates cancer

=P(have cancer and test indicate cancer)/P(est indicate cancer) =0.02*0.98/0.0392=0.5

b)

P(test does not indicate cancer)=1-P(est indicate cancer) =1-0.0392 =0.9608

hence probability of a person having cancer given that the test does not indicate cancer

=P((have cancer and test does not indicate cancer)/P(test does not indicate cancer)

=0.02*(1-0.98)/0.9608=0.0004 ~0.000

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