But it's a valid point, the question is not well-posed. If you said, "I looked at both children and saw that at least one was a girl" more people would get the right answer. Many people will assume that the author looked at only one child, not both. And there's nothing in the wording to indicate either way.
As others are pointing out, this is just the Monty Hall problem. But the way the question is posed there is much clearer.
How did you determine at least one is a girl? Presumably you looked in some way. But did you look at one child or both? That's the crux of the ambiguity.
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”.
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?
The other part doesn't change anything at all. Here you go:
Q1: I looked at only one of a pair of two randomly selected children and it was a girl. What is the probability there are two girls?
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 there are two girls?
I agree, it's perfectly clear. In my humble opinion, people are bringing their incorrect assumptions to the question, and because they're wrong, are trying to blame the framing of the question. That happens a lot with the Monty Haul paradox, as well.
And, of course, neither are paradoxes. They're just math that can seem paradoxical if you don't look closely at it.
I think it's perfectly clear that there is a straightforward and obvious interpretation of the question, which results in the answer of 2/3. There are also many strained interpretations, which result in the answer the reader wants to be right.
So if you meet someone and you are told that they have at least one girl the probability that they have a girl and a boy is 2/3, because the question has one straightforward and obvious interpretation.
And if you meet someone and you are told that they have at least one boy the probability that they have a girl and a boy is 2/3, because the question has one straightforward and obvious interpretation.
And if you meet several people and you are told that they have at least one girl the probability that they have a girl and a boy is always 2/3, because each time the question has one straightforward and obvious interpretation.
And if you meet several people and you are told that they have at least one boy the probability that they have a girl and a boy is always 2/3, because each time the question has one straightforward and obvious interpretation.
And if you meet several people and sometimes you are told that they have at least one boy and sometimes you are told that they have at least one girl the probability that they have a girl and a boy is always 2/3, because each time the question has one straightforward and obvious interpretation.
It's fine to make whatever assumption you need to get the answer you want but that doesn't make it the "straightforward and obvious interpretation". Assume your assumptions!
