This still conveys a lot of information.

Engineering is more male-heavy than lawyering.

Yes, there's more to it. The experiment was done with 70/30 and 30/70 ratios for different subjects. The book doesn't say whether they specified they were all males, my guess would be that they did.

a frequentist would take issue with the two sons, one born on a tuesday problem. you can actually count up the permutations.

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.

> 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).

Personally I wouldn't call it "Bayesian" so much as just "a conditional probability." The question doesn't ask what the probability is that a randomly selected participant is an engineer, it asks what the probability is that a participant is an engineer given that he has "typical" engineer-like traits.

But then yes, ideally you could use Bayes' Rule to find that probability.

Your assuming a US bias, the sample could have come from India which has different ratios.