Consider an autosomal recessive disease with an incidence of 1/10,000 in the general population of 100,000. Your best friend comes to you very upset because he has just taken a screening test for this disease and gotten a positive result. He is convinced he is a carrier, despite having no family history of the disease. You try to reassure him, but he says, "Don't bother. The Clinic said this test has 98% sensitivity and 90% specificity. With that level of sensitivity, it must be correct!" What is the chance your friend is NOT a carrier?
I have attempted it in this way: The reason I used Hardy-Weinberg is because there isn't any other way to determine whether he is a carrier or not.
Case 1: He is AA (double dominant). This has a 0.99^2 = 0.9801 chance. The chance that his result was positive is 1 - 0.90 = 0.1 since that is the chance that the specificity failed since specificity measures the chance of true negatives. Multiplying this gives us a 0.09801 probability of this case.
Case 2: He is Aa (heterozygous). This has a 2*0.99*0.01 = 0.0198 chance. Again his chance of a positive result should be 1 - 0.9 = 0.1. Multiplying this gives us 0.00198.
Case 3: He is aa (double recessive). This has a 0.01^2 = 0.0001 chance. The chance of getting a positive result is 0.98. Multiplying we get 0.000098.
Adding these up, I have 0.100088. We want the probability he is not a carrier, so 0.000098+0.09801 (cases 1 and 3) divided by 0.100088 (the total). This yields around 0.9802 or 98% chance, which isn't one of the choices. What am I doing wrong?