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 negative tests and 9% of positive tests prove to be incorrect. 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'? Probability-

Explanation / Answer

Here we are given that in about 26% of kidney transplants the body tries to reject the organ. Therefore,

P( reject ) = 0.26

Also, we are given that 20% of the negative tests are incorrect. Therefore,

P( negative | reject ) = 0.2 which means that P( positive | reject ) = 1 - 0.8 = 0.2

Also we know that 9% of the positive tests are incorrect. Therefore,

P( positive | not reject ) = 0.09

Probability that the test will be positive ( using the law of total probability ) , we get:

P( positive ) = P( positive | reject ) P( reject ) + P( positive | not reject ) P( not reject )

P( positive ) = 0.2*0.26 + 0.09*(1-0.26) = 0.1186

Now given that a new test has a positive result, probability that the body is atttmpting to reject the kidney is computed using the bayes theorem as:

P( reject | positive ) = P( positive | reject ) P( reject ) / P( positive) = 0.2*0.26 / 0.1186 = 0.4384

Therefore 0.4384 is the required probability here.

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