Please help: What is the probability that a person does not have cancer, given t

Question

Please help: What is the probability that a person does not have cancer, given that the PSA test is positive? (Round your answer to five decimal places.

PROBLEM: 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.

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). (DONE)

B. Suppose that 6.5% of the population has prostate cancer. What is the probability that a person does not have cancer, given that the PSA test is positive? (Round your answer to five decimal places.)

P = _______

Explanation / Answer

B. Suppose that 6.5% of the population has prostate cancer. What is the probability that a person does not have cancer, given that the PSA test is positive? (Round your answer to five decimal places.)

Pr(A person doesn't have a cancer but PSA test is positive) = ?

Pr(PSA test is positive) = Pr(Person has cancer) * Pr(Test is postive) + Pr(Person doesn't have cancer) * Pr(Test is not positive)

= 0.065 * 0.21 + 0.935 * 0.06 = 0.06975

Pr(A person doesn't have a cancer but PSA test is positive) = 0.935 * 0.06/ (0.065 * 0.21 + 0.935 * 0.06)

= 0.0561/ 0.06975 = 0.80430

so probability that person doesn't have a cancer given the test is positive = 0.80430

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