I think you are asking "how did the person who told you there is at least one girl learn that".

The answer is: it doesn't matter how because that is an unambiguous statement.

It means "you can assume the family does not have two boys".

I think people are actually getting hung up on "you are told" as if that could be a lie, or some kind of trick, when it is really just supposed to mean "here is some more information that you can rely on".

> It means "you can assume the family does not have two boys".

But it does not mean that you can assume that p(you're told at least one is a girl | both are girls) = p(you're told at least one is a girl | they aren't both girls) as explained by 6gvONxR4sf7o.

If you allow assuming whatever you want, then many answer are allowed!

That’s what it means that the problem is not “well-posed” as mentioned by in_cahoots. You need additional assumptions to get a definite answer - and the answer will depend on the assumptions.

As JeffJor noted it seems much more natural to have assumptions that keep the symmetry of the problem (because why not?) and the answer 1/2 is not just possible but arguably “better”.

These two questions are not equivalent.

Q1: I looked at only one of a pair of two randomly selected children and it was a girl. What is the probability the other I didn’t see is a girl?

Q2: I looked at both of two randomly selected children and at least one of the pair of children is a girl. What is the probability the other is also a girl?