Question

1. 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 patients that experience kidney rejection and 9% of patients that do not experience kidney rejection receive a test result that proves to be incorrect. Physicians know that in about 30% 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

Using conditional probability, it is found that there is a 0.7921 = 79.21% probability that the body is attempting to reject the kidney.

Conditional Probability

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

In which

• P(B|A) is the probability of event B happening, given that A happened.
• $P(A \cap B)$ is the probability of both A and B happening.
• P(A) is the probability of A happening.

In this problem:

• Event A: Positive test.
• Event B: Body attempting to reject the kidney.

The percentages associated with a positive test are:

• 80% of 30%(experience kidney rejection).
• 9% of 70%(do not experience kidney rejection).

Hence:

$P(A) = 0.8(0.3) + 0.09(0.7) = 0.303$

The probability of both a positive test and the body attempting to reject the kidney is:

$P(A \cap B) = 0.8(0.3) = 0.24$

Hence, the conditional probability is:

$P(B|A) = \frac{0.24}{0.303} = 0.7921$

0.7921 = 79.21% probability that the body is attempting to reject the kidney.

A similar problem is given at https://brainly.com/question/25553471