Question

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 0.02 probability that a male patient has prostate cancer before testing. the probability of a false - positive test is 0.75, and the probability of a false - negative (no indication of cancer when cancer is actually present) is 0.20. let $c$ = event male 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 males, the prior probability of having cancer increases. suppose that the prior probability of the male patient is 0.3 rather than 0.02. 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 <select your answer> than the difference between $p(c|+)$ and $p(c| - )$ in part (c).

Explanation:

Step1: Identify given probabilities

Let $P(C)=0.3$ (prior probability of having cancer), $P(+|C^c)=0.75$ (false - positive rate), $P(-|C)=0.20$ (false - negative rate). Then $P(-|C^c)=1 - 0.75 = 0.25$ and $P(+|C)=1 - 0.20 = 0.80$.

Step2: Calculate $P(+)$ using the law of total probability

$P(+)=P(+|C)P(C)+P(+|C^c)P(C^c)=0.80\times0.3 + 0.75\times(1 - 0.3)=0.80\times0.3+0.75\times0.7=0.24 + 0.525 = 0.765$.

Step3: Calculate $P(C|+)$ using Bayes' theorem

$P(C|+)=\frac{P(+|C)P(C)}{P(+)}=\frac{0.80\times0.3}{0.765}=\frac{0.24}{0.765}\approx0.3137$.

Step4: Calculate $P(-)$ using the law of total probability

$P(-)=P(-|C)P(C)+P(-|C^c)P(C^c)=0.20\times0.3+0.25\times(1 - 0.3)=0.20\times0.3 + 0.25\times0.7=0.06+0.175 = 0.235$.

Step5: Calculate $P(C| -)$ using Bayes' theorem

$P(C| -)=\frac{P(-|C)P(C)}{P(-)}=\frac{0.20\times0.3}{0.235}=\frac{0.06}{0.235}\approx0.2553$.

Answer:

c. Positive test: 0.3137
Negative test: 0.2553
d. The difference between $P(C|+)$ and $P(C|-)$ in parts (a) and (b) is smaller than the difference between $P(C|+)$ and $P(C|-)$ in part (c).