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 othermeans) that 3% 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 97% 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? (Round to three decimal places as needed.)

What is the probability of a person having cancer given that the test does not indicate cancer? (Round to six decimal places as needed.)

Answer #1

Probability of a randomly chosen person having cancer given that the test indicates cancer =

Total people having cancer = 0.03*2000 = 60

Total people who are correctly shown by device having cancer = 0.97*60 = 58.2

Total people who are incorrectly shown by device that they are having cancer = 0.97*2000*0.01 = 19.4

Required Probability =58.2 / (58.2+19.4) = 0.75

Probability of a person having cancer given that the test does not indicate cancer=

Total people who are having cancer but the test does not indicate = 60-58.2 =1.8

Total number of people who do not have cancer and the test does not indicate cancer = 0.97*2000*0.99 = 1920.6

Required Probability = 1.8 /(1.8+1920.6) = 0.00093 = 0.001

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