Prostate Cancer Test
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According to a 2018 article in Esquire magazine, approximately 70% of males over age 70 will develop cancerous cells in their prostate. Prostate cancer is second only to skin cancer as the most common form of cancer for males in the United States. One of the most common tests for the detection of prostate cancer is the prostate-specific antigen (PSA) test. However, this test is known to have a high false-positive rate (tests that come back positive for cancer when no cancer is present).
Suppose there is a .02 probability that a male patient has prostate cancer before testing. The probability of a false-positive test is .75, and the probability of a false-negative (no indication of cancer when cancer is actually present) is .20.
a) What is the probability that the male patient has prostate cancer if the PSA test comes back positive?
b) What is the probability that the male patient has prostate cancer if the PSA test comes back negative?
For older men, the prior probability of having cancer increases.
c) Suppose that the prior probability of the male patient is .3 rather than .02. What is the probability that the male patient has prostate cancer if the PSA test comes back positive? What is the probability that the male patient has prostate cancer if the PSA test comes back negative?
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Solution Summary
The solution includes step by step calculations to create a table for the problem, as well as calculations using Bayes' theorem.
Solution Preview
A table is developed to apply Bayes' Theorem to answer the questions.
The calculation for each box in the table is shown.
Has cancer No cancer Total
Test pos (Step 7) 1.6 (Step 4) 73.5 (Step 8) 75.1
Test neg (Step 5) 0.4 (Step 6) 24.5 (Step 9) 24.9
Total (Step 2) 2 (Step 3) 98 (Step 1) 100
The total is 100.
(Step 1) = 100
There is a .02 probability that a male patient has prostate cancer before testing.
(Step 2) = 0.02 * (Step 1) = 0.02 * 100 = 2
(Step 3) = (Step 1) - (Step 2) ...
Education
- MSc, California State Polytechnic University, Pomona
- MBA, University of California, Riverside
- BSc, California State Polytechnic University, Pomona
- BSc, California State Polytechnic University, Pomona
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