(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.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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