This is probably the wrong place for this comment. However, this seems to be the most recent thread associated with "An Intuitive Explanation of Bayes's Theorem", so I think this is the least bad location. Unfortunately I may wind up doing this sort of thing a lot, though, as I work my way through The Sequences.

It's a fantastic piece of work, but I have a couple comments and corrections for general discussion. Maybe they are technicalities and aren't worth the trouble of changing, or maybe some are worthwhile.

1) In the egg/pearl problems, it isn’t explicitly stated that all the eggs are either painted red or blue. Similarly, it is possible, at least in the later problems, that there are some other objects in the eggs besides pearls. This means that p(pearl) + p(empty) < 1. This is not an issue in the breast cancer problems, because the reader can safely assume that a patient either has breast cancer or does not; there is no third state. A reader trained in medicine might, however, be inclined to speculate that a mammography test could also come out “inconclusive” or some such third state besides just “positive” and “negative”.

2) The section dealing with the hypothetical “Tams-Braylor” breast cancer test and likelihood ratios doesn’t explicitly show how to how to compute posterior probabilities using only the likelihood ratio and the initial probability. This new equation can be derived in a single step, but the new form of the equation is not as intuitive for me as the original, so I personally had to derive it:

p(cancer|positive) = 0.720.01/(0.720.01+[1-0.01]*0.0048)

This says that the chances of cancer among women who test positive is the % with cancer divided by the total % who test positive (that is to say, the true positives plus the false positives).

p(cancer|positive) = [0.72/0.0048]0.01/([0.72/0.0048]0.01+1-0.01)

This new equation says that the chances of cancer among women who test positive is the % cancer detections per false positive rate divided by the total % who test positive per false positive (that is to say, % cancer detections per false positive rate plus the % false positives).

Only when looking at it afterward did it start to become intuitive, even though I was able to solve all of the previous problems just by thinking about what made sense rather than plugging anything into any equation or algorithm. (Including solving the problem, using your Javascript calculator, that you said “You probably shouldn't try to solve this with just a Javascript calculator, though.”) Although forcing the reader to do a little extra thinking may be good for those who have a technical background and will be able to fill in the missing gaps, the majority of your target audience may not be confident enough in their newfound knowledge to do so. Perhaps you could get the best of both worlds by adding a small paragraph after giving the solution?

3) Perhaps it would also be useful to explicitly point out how understanding various aspects of Bayes’ theorem should prove useful in everyday life, rather than giving the reader the impression that it is merely analogous to everyday problems. This would just be a few sentences scattered throughout.

4) There isn’t a Javascript calculator for the red chip/blue chip in a backpack problem.