Question

A new 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 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.
a) Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer?
b) What is the probability of a person having cancer given that the test does not indicate cancer?

Answer 1

The tree diagram of the problem above is attached
There are four outcomes of the two events,

First test - Cancer, Second Test - Cancer, the probability is 0.0396
First test - Cancer, Second Test - No Cancer, the probability is 0.0004
First test -  No Cancer, Second Test - There is cancer, the probability is 0.0096
First test - No cancer, Second Test - No cancer, the probability is 0.9054

The probability of someone picked at random has cancer given that test result indicates cancer is  $\frac{0.0396}{0.0396+0.0096}= \frac{33}{41}$

The probability of someone picked at random has cancer given that test result indicates no cancer is $\frac{0.0396}{0.0004+0.9504} = \frac{99}{2377}$