This still conveys a lot of information.
Engineering is more male-heavy than lawyering.
Engineering is more male-heavy than lawyering.
let's say we have 10 engineers, 9 of them are male. we also have 10 lawyers, 6 of them are male. Let's say one in 10 people likes doing math on the weekend.
There are 90 out of 100 ways to have a group of male engineers, one of which who likes math, but only 60 out of 100 ways to do the same with a male lawyer. furthermore, if we add in the four kids as another 1 in 10 thing, the situation gets even worse. This isn't bayesian, this is just counting boxes on a permutation table.
It's the same thing. Bayes' theorem allows you to shortcut straight to the answer without having to draw out a full probability tree / permutation table. But the underlying math is the same - in each case you have a different probability of B given A, versus B given (not A).
But then yes, ideally you could use Bayes' Rule to find that probability.