Transplant operations have become routine. One common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20% of the rejections will incorrectly receive negative test results and 9% of the kidneys that are accepted will receive incorrect positive test results. Physicians know that in about 29% of kidney transplants the body tries to reject the organ.

Question

Transplant operations have become routine. One common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20% of the rejections will incorrectly receive negative test results and 9% of the kidneys that are accepted will receive incorrect positive test results. Physicians know that in about 29% of kidney transplants the body tries to reject the organ.

If the new test has a positive result (indicating early warning of rejection), what is the probability that the body is attempting to reject the kidney?

Answer 1

To determine the probability that the body is attempting to reject the kidney when the new urine test has a positive result, we can use Bayes' theorem.

Let's define the following probabilities:
A: Body attempting to reject the kidney
B: Positive test result

We are given:
P(A) = 0.29 (probability that the body tries to reject the organ)
P(B|A) = 0.91 (probability of a positive test result given that the body is attempting to reject the kidney)
P(B|not A) = 0.09 (probability of a positive test result given that the body is not attempting to reject the kidney)

We want to find:
P(A|B) (probability that the body is attempting to reject the kidney given a positive test result)

Using Bayes' theorem:
P(A|B) = (P(A) * P(B|A)) / P(B)

To find P(B), we need to consider both true positives (positive test result when the body is attempting to reject) and false positives (positive test result when the body is not attempting to reject):
P(B) = P(A) * P(B|A) + P(not A) * P(B|not A)

P(not A) = 1 - P(A) = 1 - 0.29 = 0.71 (probability that the body is not attempting to reject the organ)

Substituting the values into the formula:
P(B) = 0.29 * 0.91 + 0.71 * 0.09
P(B) = 0.2639 + 0.0639
P(B) = 0.3278

Now, we can calculate P(A|B):
P(A|B) = (P(A) * P(B|A)) / P(B)
P(A|B) = (0.29 * 0.91) / 0.3278
P(A|B) = 0.2639 / 0.3278
P(A|B) ≈ 0.8043 or approximately 80.43%

Therefore, if the new test has a positive result, there is approximately an 80.43% probability that the body is attempting to reject the kidney.

12

To determine the probability that the body is attempting to reject the kidney when the new urine test has a positive result, we can use Bayes' theorem.