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96 out of 1000 women without breast cancer will also get positive mammographies.

If 1000 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?

950 out of 9,900 women without breast cancer will also get a positive mammography.

The intent is to convey, not abstract rules for manipulating numbers, but what the numbers mean, and why the rules are what they are (and cannot possibly be anything else).This is the correct answer, the answer a doctor should give a positive-mammography patient if she asks about the chance she has breast cancer; if thirteen patients ask this question, roughly 1 out of those 13 will have cancer.The most common mistake is to ignore the original fraction of women with breast cancer, and the fraction of women without breast cancer who receive false positives, and focus only on the fraction of women with breast cancer who get positive results. Maybe you understand it in theory, but every time you try to apply it in practice you get mixed up trying to remember the difference between belongs in the numerator or the denominator.Why does a mathematical concept generate this strange enthusiasm in its students? While there are a few existing online explanations of Bayes' Theorem, my experience with trying to introduce people to Bayesian reasoning is that the existing online explanations are too abstract.

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