P(G@M|I & W) < P(G@M| I & M)

where we (somewhat simplisticly) represent the outcomes as Yes/No deciscions (same idea if they are continous measures of ability but more complicated)

G@M: The person is good at math
W: The person is a woman
M: The Person is a man
I: The person has whatever extra information you have about the person in the context you wish to consider. In a grad school applcation process this would be something like has GRE score such and such, has college grades this and that, has publications etc..

All Bayes theorem would let us do in this case is rearange the inequality to say

P(I& W|G@M)/P(I&W} < P(I& M|G@M)/P(I&M}

You can look at this as saying that you have reason to suspect women with the same test scores/grades as men are worse than men if and only if women who are good at math compose a smaller percentage of people who score that well on the tests/get grades like that than me who are good at math compoe of men who get test scores.

But that doesn’t really add any information. I don’t really see how Bayes theorem is that helpful here.

And because I’m continually trying to really internalize Bayes’ Rule, where precisely might it apply here?