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The correct answer is 7.8%, obtained as follows: Out of 10,000 women, 100 have breast cancer; 80 of those 100 have positive mammographies.
From the same 10,000 women, 9,900 will not have breast cancer and of those 9,900 women, 950 will also get positive mammographies.
9.6% of women without breast cancer will also get positive mammographies.
800 out of 1000 women with breast cancer will get positive mammographies. 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.To see that the final answer always depends on the original fraction of women with breast cancer, consider an alternate universe in which only one woman out of a million has breast cancer.Even if mammography in this world detects breast cancer in 8 out of 10 cases, while returning a false positive on a woman without breast cancer in only 1 out of 10 cases, there will still be a hundred thousand false positives for every real case of cancer detected.
This makes the total number of women with positive mammographies 950 80 or 1,030.