[dalmo3]

I immediately modelled the problem like you did, then I thought of this interesting variation:

"I have two children, Michael and Alex. Michael is a boy. What's the probability of both being boys?"

If you make a truth table with names as columns, you clearly have only two possibilities for Michael=1.

However if you pick older/younger again you're back to 3 possible states.

I think the answer is still 1/3, but it's a trickier one to reason about immediately.

It seems the question adds information by naming the children, but there's a hidden statement in the form "at least one of them is Michael", which invalidates a truth table with names as columns.

I can only conclude that birth order is an underlying property of the entity. A strict, real differentiator as much as sex is. Names aren't, so names don't add information in this case.

Is there a term for that? Or am I just wrong?

[kgwgk]

In the original problem you start by assuming 4 equally-probable cases Bb Bg Gb Gg [1] and you ask a question. Depending on the answer you are in the Bb/Bg/Gb or the Gg subsets.

In your variant you need additional assumptions. Will the person always tell you the sex and name of the eldest? Or the names of the boys?

“Michael is a boy” is not really different from “the youngest is a boy”. The probability of both being boys depends on why are you being told that.

[1] Depending in the context the assumption may not be appropriate (a extreme example may be China).