This seems to me to be the best way to understand Bayes' theorem, with natural frequencies being the best way to mentally compute with it. E.g. for the breast cancer example,<p>- Imagine there are 1000 women who participate in routine screening<p>- 1% → 10 of these have breast cancer<p>- 80% → 8 of these will get positive mammograms<p>- 990 don't have breast cancer<p>- 9.6% ≅ 10% → 99 of the 990 get positive mammograms<p>So that the probability of having breast cancer, given a positive mammogram, is ≅ 8/99 ≅ 8%.<p>There is a bunch of research on natural frequencies being generally the best way to reason about this sort of thing.