Question

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)? 0.0213 b. what is the probability that the male patient has prostate cancer if the psa test comes back negative (to 4 decimals)? 0.0161 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)? 0.3137 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 lower than the difference between (p(c|+)) and (p(c|-)) in part (c).

Explanation:

Step1: Define probabilities

Let $P(C)=0.3$ (new prior probability of having cancer), $P(+|
eg C)=0.75$ (false - positive rate), $P(-|C)=0.20$ (false - negative rate). Then $P(
eg C)=1 - P(C)=0.7$, $P(+|C)=1 - P(-|C)=0.8$.

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

$P(+)=P(+|C)P(C)+P(+|
eg C)P(
eg C)=0.8\times0.3 + 0.75\times0.7=0.24+0.525 = 0.765$.

Step3: Calculate $P(-)$

$P(-)=1 - P(+)=1 - 0.765 = 0.235$.

Step4: Calculate $P(C| -)$ using Bayes' theorem

$P(C| -)=\frac{P(-|C)P(C)}{P(-)}=\frac{0.2\times0.3}{0.235}=\frac{0.06}{0.235}\approx0.2553$.

Answer:

0.2553