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The question goes like this: The American Diabetes Association estimates that 5.9% of Americans have diabetes. Suppose that a medical lab has developed a simple diagnostic test for diabetes that is 98% accurate for people who have the disease and 95% accurate for people who do not have it. If the medical lab gives the test to a randomly selected person, what is the probability that the person has diabetes given a positive test?

I'm not exactly sure where to start, but I was thinking about the 98% accurate test of positive for those with diabetes and the 5% inaccuracy of the test stating that you have a positive test when you don't. I do note however that 94.1% of the population don't have diabetes whereas 5.9% do have it, which means that there is a greater stress on the inaccurate test of positive for those that don't have the disease.

Bluedragon01313
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  • $P(D|T) = P(D\cap T)/P(T) = P(D)P(T|D) = (.059)(.98)/P(T),$ where $D$ means has disease; $T$ means tests positive, and $P(T) = P(D\cap T) + P(D^c \cap T)$ by the Law of Total Probability. You already have $P(D\cap T),$ now use other given info along with the complement rule to find $P(D^c \cap T). $ This is a straightforward application of Bayes' Theorem with several 'Related' links in the right margin of this page pointing to similar problems/solutions. – BruceET Mar 15 '19 at 21:14
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    https://en.m.wikipedia.org/wiki/Bayes'_theorem#Drug_testing – amd Mar 15 '19 at 21:28
  • @amd. Certainly similar, but different numbers for prevalence, sensitivity, and specificity. Don't know whether that counts as a 'Duplicate'. – BruceET Mar 15 '19 at 22:27
  • @BruceET It does in my mind. Aside from the numbers to be plugged into the formulas, they are asking exactly the same thing, which moreover is exactly the same problem as described in the Wikipedia article, albeit with drug testing instead of diagnosis.. Would you really consider “Find the intersection of $2x-3y=1$ and $x+y=2$” and “find the intersection of $x+5y=0$ and $3x-y=2$” to be different questions? Or “How do I compute the null space of matrix $A$” for two different matrices? – amd Mar 15 '19 at 23:20
  • @amd. I have never considered myself an expert on deciding about duplicates. (If it's exactly a second copy of a textbook exercise, then I am sure.) Why not try flagging it as a duplicate and see if you get agreement. – BruceET Mar 15 '19 at 23:25
  • Possibly helpful: https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 – Ethan Bolker Mar 16 '19 at 01:43
  • Related: https://math.stackexchange.com/questions/32933/describing-bayesian-probability – Henry Oct 08 '20 at 10:08

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If the test for those without diabetes has 95% accuracy then 100 - 95 = 5% that may have diabetes with the 2nd test.

If the 1st test that is 95% accurate AND the 2nd test is also given, assuming that the tests are different then taking the 5.9% from the earlier sentence I reckon that 5.9% X 95% X 5% = 0.059 X 0.95 X 0.05 = 0.0028025

or approximately 0.0028 very approximately 3 people per 1000 will be positive for diabetes.

However this statement seems a little contradictory "Suppose that a medical lab has developed a simple diagnostic test for diabetes that is 98% accurate for people who have the disease and 95% accurate for people who do not have it" as I thought there was only one test for diabetes unless one is a test for TYPE 1 diabetes and the other a test of TYPE 2 diabetes. So my answer is based on the assumption of two separate tests.

John Anthony Oliver
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