What is the probability that the body is going to reject the kidney?
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 has 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. When the test is conducted on someone whose kidney will be rejected, approximately one out of five tests will be negative(i.e, the test is wrong). When the test is conducted on a person whose kidney will not be rejected, 8% will show a positive test result(i.e., another incorrect result). Physicians know that in about 35% of kidney transplants the body tries to reject the organ.Suppose that the test was performed and the test is positive(indicating early warning of rejection).
Physicians know that in 35% of kidney transplants the body rejects the kidney. In 65% of transplants the body does not reject the kidney. Assume the test is performed on a person whose kidney is going to be rejected. There is a (4/5) probability that the test is positive. The probability that the body is going to reject the kidney due to the test showing positive is 0.35*(4/5). When the test is conducted on a person whose kidney is not going to be rejected there is a 8% probability that the test is positive.
If the test shows positive the probability that the body is going to reject the kidney is given by `(0.35*0.8)/(0.35*0.8 + 0.65*0.08)` = `0.28/(0.28 + 0.052)`= 70/83
The probability that the body is going to reject the kidney if the test shows positive is equal to 70/83.