Question 872645: Around 5.8% of the people living in England are thought to be diabetic. A pharmaceutical company has developed a new test for identifying whether patients are diabetic, and found during clinical trials that • if the person is diabetic, the test comes back positive 87% of the time • if the person isn’t diabetic, the test comes back negative 96% of the time
a) Suppose a randomly selected person in England is tested for diabetes using this new test. If the test comes back positive, what is the probability that the person is diabetic?
b) Suppose John and Mark both live in England, and that precisely one of them is diabetic. If John is tested for diabetes using the new test and the test comes back positive, what is the probability that Mark is diabetic
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Of the 5.8% who are diabetic, 87% get a positive test. Those people who are diabetic and get a positive test are 87% of 5.8% of the population.
That is (87%)(5.8%) = 5.046%.
The rest of the diabetics, accounting for 5.8% - 5.046% = 0.754% of the total population, get a negative test.
Among the 100% - 5.8% = 94.2% of the population who are not diabetic, 4% get a positive test.
That means that (4%)(94.2%) = 3.768% of the population will be needlessly worried about their positive diabetes test.
All in all, 5.046% + 3.768% = 8.814% of the population got a positive test for diabetes.
a) The fraction of people with positive tests who are truly diabetic is
So if a randomly selected person gets tested and the test is positive, the probability that he/she is diabetic is 57.25%.
b) If John is tested for diabetes using the new test and the test comes back positive, it is 57.25% probable that John is diabetic, but it is
100% - 57.25% = 42.75% probable that John is not diabetic.
So, if John is tested for diabetes using the new test and the test comes back positive, it is 42.75% probable that John is not diabetic, but Mark is diabetic.
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