Question

The prostate-specific antigen (PSA) test is a simple blood test to screen for prostate cancer. It has been used in men over 50 as a routine part of a physical exam, with levels above 4 ng/mL indicating possible prostate cancer. The test result is not always correct, sometimes indicating prostate cancer when it is not present and often missing prostate cancer that is present. Suppose that these are the approximate conditional probabilities of a positive (above 4 ng/ml) and negative test result given cancer is present or absent.

Test Result
Positive Negative
Cancer present 0.21 0.79
Cancer absent 0.06 0.94
In a large study of prostate cancer screening, it was found that about 6.5% of the population has prostate cancer.
A. Draw a tree diagram for selecting a person from this population (outcomes: cancer present or absent) and testing his blood (outcomes: test positive or negative).
B. What is the probability that the test is positive for a randomly chosen person from this population?

Answer 1

Explanation:

A tree diagram is the right way of determining a list of sample elements in the sample space. It is by finding the ordered n-tuples of the elements in the events under consideration.

We can found the tree diagram for selecting a person from the population in the given question in the attached image below.

b. Given that 6.5% of the population have prostate cancer. Suppose the total population is 100000. Then, we have the following table:

By applying Naive-Bayes theorem, the probability that the test is positive for a randomly chosen person from this population is expressed as:

P( positive ) = P( positive | cancer present ) P( cancer present ) + P( positive | cancer absent ) P( cancer absent )

P( positive ) = 0.21×0.065 + 0.06×(1 - 0.065)

P( positive ) = 0.01365 + 0.0561

P( positive ) = 0.06975

Thus, the probability that the test is positive for a randomly chosen person from this population is:

= 0.06975