Either way, the article's explanation for that particular question is wrong.
"Dick is a 30-year-old man. He is married with no children. A man of high ability and high motivation, he promises to be quite successful in his field. He is well liked by his colleagues.
This description was intended to convey no information relevant to the question of whether Dick is an engineer or a lawyer."
The description in the quiz is very different.
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.