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. What is the probability that the test is positive for a randomly chosen person from this population?

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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.

                                    Positive Result                 Negative Result

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 6.6% of the population has prostate cancer.

What is the probability that the test is positive for a randomly chosen person from this population? (Enter your answer to five decimal places.)

P(Positive test) =

P(Positive Result) = 0.0699

Step-by-step explanation:

We are given that 6.6% of the population has prostate cancer. So,

P(Cancer Present) = 0.066

P(Cancer Absent) = 1 - 0.066 = 0.934

From the given conditional probability table, we have:

P(Positive Result | Cancer Present) = 0.21

P(Positive Result | Cancer Absent) = 0.06

We need to find the probability that the result is positive. It can be either that the result is positive and cancer is present or the result is positive and the cancer is absent. So,

P(Positive Result) = P(Positive Result∩Cancer Present) + P(Positive Result∩Cancer Absent)

                             = P(Positive Result | Cancer Present)*P(Cancer Present) + P(Positive Result | Cancer Absent)*P(Cancer Absent)

                             = (0.21)*(0.066) + (0.06)*(0.934)

                             = 0.01386 + 0.05604

P(Positive Result) = 0.0699

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