Exercise $04\mathrm{.}43($Bayes Theorem$)$

Question $12$ of $17$

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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 $0\mathrm{.}02$ probability that a male patient has prostate cancer before testing. The probability of a false$-$positive test is $0\mathrm{.}75,$ and the probability of a false$-$negative $($no indication of cancer when cancer is actually present$)$ is $0\mathrm{.}20.$

Let $C=$ event nale patient has prostate cancer

$+=$ positive PSA test for prostate cancer

$-=$ negative PSA test for prostate cancer

a$.$ What is the probability that the male patient has prostate cancer if the PSA test comes back positive $($to $4$ decimals$)?$

b$.$ What is the probability that the male patient has prostate cancer if the PSA test comes back negative $($to $4$ decimals$)?$

c$.$ For older men, the prior probability of having cancer increases. Suppose that the prior probability of the male patient is $0\mathrm{.}3$ rather than $0\mathrm{.}02.$ What is the probability that the male patient has prostate cancer if the PSA test comes back positive $($to $4$ decimals$)?$

What is the probability that the male patient has prostate cancer if the PSA test comes back negative $($to $4$ decimals$)?$

d$.$ What can you infer about the PSA test from the results of parts $($a$),($b$),$ and $($c$)?$

The difference between $P(C|+)$ and $P(C|-)$ in parts $(a)$ and $(b)$ is than the difference between $P(C|+)$ and $P(C|-)$ in part $($c$).$

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