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Section A.1 Activity on Medical Testing

A naturally occuring random genetic mutation called M is found at a rate of \(1\) in \(100,000\) people. A test called T detects mutation M with \(95\%\) accuracy. This means that \(5\%\) of people with mutation M will test negative, and \(5\%\) of people who do not have mutation M will test positive. As part of a medical study with randomly selected participants, Skylar takes test T and the result is positive (that is, the test indicates that Skylar has mutation M). Before getting the positive test result, the best estimate of the chance that Skylar has mutation M is \(1\) in \(100,000\) , which is \(0.00001=0.001\%\text{.}\) How does the positive test result change this estimate?
Problem : Estimate the chance that Skylar has mutation M, knowing that Skylar tested positive for mutation M.
To solve the problem, consider what would happen if everyone in a population of \(100\) million people got the test for mutation M? Let’s name these subgroups of the population.
  • \(A\) : people who have mutation M
  • \(B\) : people who do not have mutation M
  • \(P\) : people whose test result is positive
  • \(N\) : people whose test result is negative
It will be helpful to consider further subdivisions of these groups of people.
  • \(AP\) : people in group A who test positive
  • \(AN\) : people in group A who test negative
  • \(BP\) : people in group B who test positive
  • \(BN\) : people in group B who test negative
Solve the problem.
  1. Estimate how many people are in each group. Hint: start with groups \(A\) and \(B\) , then do \(AP\) , \(AN\) , \(BP\) , \(BN\) , then do \(P\) and \(N\) last.
  2. Now solve the Problem using the numbers from part 1. Give your solution as a percent, and write a sentence or two to express your reasoning.
  3. Would your answer change if you thought about a population of \(10\) million instead of \(100\) million?
  4. Would your answer change if mutation M occurs in \(10\%\) of the population, instead of \(1\) in \(100,000\) ?