There is a screening test for prostate cancer that looks at the level of PSA (prostate-specific antigen) in the blood. There are a number of reasons besides prostate cancer that a man can have elevated PSA levels. In addition, many types of prostate cancer develop so slowly that they are never a problem. Unfortunately, there is currently no test to distinguish the different types, and using the test is controversial because it is hard to quantify the accuracy rates and the harm done by false positives. For this problem, we’ll call a positive test a true positive if it catches a dangerous type of prostate cancer. We’ll assume the following numbers: The rate of prostate cancer among men over 50 = 0.0005 The true positive rate for the test = 0.9 The false positive rate for the test = 0.01 Let 𝑇 be the event a man has a positive test and let 𝐷 be the event a man has a dangerous type of the disease. Find 𝑃(𝐷|𝑇 ) and 𝑃(𝐷|𝑇𝑐).
Question
There is a screening test for prostate cancer that looks at the level of PSA (prostate-specific antigen) in the blood. There are a number of reasons besides prostate cancer that a man can have elevated PSA levels. In addition, many types of prostate cancer develop so slowly that they are never a problem. Unfortunately, there is currently no test to distinguish the different types, and using the test is controversial because it is hard to quantify the accuracy rates and the harm done by false positives. For this problem, we’ll call a positive test a true positive if it catches a dangerous type of prostate cancer. We’ll assume the following numbers: The rate of prostate cancer among men over 50 = 0.0005 The true positive rate for the test = 0.9 The false positive rate for the test = 0.01 Let 𝑇 be the event a man has a positive test and let 𝐷 be the event a man has a dangerous type of the disease. Find 𝑃(𝐷|𝑇 ) and 𝑃(𝐷|𝑇𝑐).
Solution
To solve this problem, we need to use Bayes' theorem, which is a fundamental theorem in the field of probability and statistics that describes how to update the probabilities of hypotheses when given evidence.
Bayes' theorem is usually stated as follows:
P(A|B) = [P(B|A) * P(A)] / P(B)
In this case, we want to find P(D|T) and P(D|Tc), where:
- D is the event a man has a dangerous type of the disease (prostate cancer).
- T is the event a man has a positive test.
- Tc is the event a man has a negative test (the complement of T).
From the problem, we know that:
- P(D) = 0.0005 (the rate of prostate cancer among men over 50)
- P(T|D) = 0.9 (the true positive rate for the test)
- P(T|Dc) = 0.01 (the false positive rate for the test)
First, let's find P(T), the probability of a positive test. This is the sum of the probability of a true positive and the probability of a false positive:
P(T) = P(T and D) + P(T and Dc) = P(T|D) * P(D) + P(T|Dc) * P(Dc) = 0.9 * 0.0005 + 0.01 * (1 - 0.0005) = 0.00045 + 0.009995 = 0.010445
Now we can find P(D|T) using Bayes' theorem:
P(D|T) = [P(T|D) * P(D)] / P(T) = (0.9 * 0.0005) / 0.010445 = 0.0431
This means that if a man has a positive test, there is a 4.31% chance that he has a dangerous type of prostate cancer.
Next, let's find P(D|Tc). Since a negative test result and having the disease are mutually exclusive events, P(D|Tc) is 0.
So, if a man has a negative test, there is a 0% chance that he has a dangerous type of prostate cancer.
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