Question 872645
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
{{{"5.046 %"/"8.814 %"=0.5725= "57.25 %"}}}
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.