Question

According to a 2018 article in Esquire magazine, approximately 70% of males over age 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)?
0.0214 [I did this part correctly but I had trouble finding the other parts, they kept marking them incorrectly]

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 0.3 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)?

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

Answer 1

Answer:

a) Probability of prostrate cancer given a positive test is P(C|+) = 0.0213

b) Probability of cancer given a negative test is P(C|-) = 0.0161

c) Probability of prostrate cancer given a positive test is P(C|+) = 0.3137

d) Probability of cancer given a negative test is P(C|-) = 0.2553

Explanation:

Probability male patient has prostate cancer, P(C) = 0.02

Probability male patient does not have prostrate cancer P(C') = 1 - 0.02 = 0.98

Probability of a positive test given there is no cancer, i.e. P(false positive) = P(+|C') = 0.75

P(negative test given there is cancer) = P(false negative) = P(-|C) = 0.2

P(negative test given there is no cancer) is the complement of P(+|C') = P(-|C') = 1 - 0.75 = 0.25

Probability of positive test given there is prostrate cancer, P(+|C) is the complement of P(-|C), = 1 - 0.2 = 0.8.

a) Probability of prostrate cancer given a positive test is P(C|+)

According to Baye's theorem, P(C|+) = P(+|C)P(C)/P(+)

For P(+), we use the Law Of Total Probability: P(+) = P(+|C)P(C) + P(+|C')P(C')

P(+) = (0.8 * 0.02) + (0.75 * 0.98) = 0.751

Therefore, P(C|+) = P(+|C)P(C)/P(+)

P(C|+) = (0.8 * 0.02)/0.751 = 0.0213

b) Probability of cancer given a negative test is P(C|-)

According to Baye's theorem, P(C|-) = P(-|C)P(C)/P(-)

P(-) = P(-|C)P(C) + P(-|C')P(C')

P(-) = (0.2 * 0.02) + (0.25 * 0.98) = 0.249

Therefore, P(C|-) = (0.2 * 0.02)/0.249

P(C|-) = 0.0161

Part 2: Given the following;

Probability male patient has prostate cancer, P(C) = 0.3

Probability male patient does not have prostrate cancer P(C') = 1 - 0.3 = 0.70

Probability of a positive test given there is no cancer, i.e. P(false positive) = P(+|C') = = 0.75

P(negative test given there is cancer) = P(false negative) = P(-|C) = 0.2

P(negative test given there is no cancer) is the complement of P(+|C') = P(-|C') = 1 - 0.75 = 0.25

Probability of positive test given there is prostrate cancer, P(+|C) is the complement of P(-|C), = 1 - 0.2 = 0.8.

c) Probability of prostrate cancer given a positive test is P(C|+)

According to Baye's theorem, P(C|+) = P(+|C)P(C)/P(+)

For P(+), we use the Law Of Total Probability: P(+) = P(+|C)P(C) + P(+|C')P(C')

P(+) = (0.8 * 0.3) + (0.75 * 0.7) = 0.751

Therefore, P(C|+) = P(+|C)P(C)/P(+)

P(C|+) = (0.8 * 0.3)/0.765 = 0.3137

d) Probability of cancer given a negative test is P(C|-)

According to Baye's theorem, P(C|-) = P(-|C)P(C)/P(-)

P(-) = P(-|C)P(C) + P(-|C')P(C')

P(-) = (0.2 * 0.3) + (0.25 * 0.7) = 0.235

Therefore, P(C|-) = (0.2 * 0.3)/0.235

P(C|-) = 0.2553