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Question about Bayesian updates.
Say Jane goes to get a cancer screening. 5% prior of having cancer, the machine has a success rate of 80% and a false positive rate of 9%. Jane gets a positive on the test and so she now has a ~30% chance of having cancer.
Jane goes to get a second opinion across the country. A second cancer screening (same success/false positive rates) says she doesn't have cancer. What is her probability for having cancer now?
(Assuming that two tests are independent, which is a rather unrealistic assumption in this case) If you know how to calculate the ~30% answer to the first part of the question, then this problem is pretty straightforward to solve. Just use Bayes' rule again, treating the posterior from your first calculation (~30%) as your prior for the next calculation.
If Kim came from a population that had a ~30% prior of having cancer and took one test which came out negative, then her probability after that one test would be the same as Jane's probability after both tests.