Transplant operations have become routine. One common transplant operation is fo

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 5% of the kidneys that are accepted will receive incorrect positive test results. Physicians know that in about 26% 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? Express your answer rounded correctly to three decimal places. Probability =

Explanation / Answer

Solution:- R = body trying to reject kidney
+ve = test indicates a positive result

i shall initially use percentage probabilities to avoid decimals

P[R & +ve] = 26% * 80% = 20.80%
P[R' & +ve] = 74% * 5% = 3.70%
P[+ve] = sum of the above = 20.80% + 3.70% = 24.50%

P[R | +ve] = P[R & +ve] / P[+ve]
= 20.8/24.50
= 0.849

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