In the correct version of this story, the mathematician says "I have two children", and you ask, "Is at least one a boy?", and she answers "Yes". Then the probability is 1/3 that they are both boys.
If I told someone that in real life and they replied by asking, "Is at least one a boy?", I would probably back away very, very slowly.
The main point is that given the way Atwood asked the question the correct answer is 50% which wasn't the answer he intended to discuss and indicates he himself didn't fully understand the subject he was writing about. It wouldn't be the first time either.
Atwood doesn't seem wrong or particularly unclear to me. If you "met someone who told you they had two children, and one of them is a girl", then presumably we should imagine this person saying: "I have two children and one of them is a girl," and not "I have two children, X and Y, and X is a girl." Obviously in the first case, we don't know if it's X or Y that's the girl, so the set of possible worlds is [(X-G & Y-G), (X-G & Y-B), (X-B & Y-G)], so we get the 2/3 answer. But maybe other people don't share my linguistic intuitions...
If I met someone who told me "I have two children, and one of them is a girl," I would be pretty sure they have one girl and one boy. If they had two girls, they would have said "I have two children, and both of them are girls." So the English statement has some implicit information. On the other hand, if I were given a riddle with that statement, I would know the implicit information may be intentionally misleading. In this case, as Paul Buchheit says, the statement does not have enough information.
Yes, the question must be very carefully worded (which is kind of my point, if I have one).
In order to get the "unintuitive" outcome, there must be some element of selection (much like the Monty Hall problem has). In your formulation of the question, the GG possibility has already been eliminated because the mathematician answers "yes".
What do you think of this story? I get a frantic call from my friend at the airport. He's been captured by the TSA for bringing a nose hair trimmer onto the plane. He was taking his brothers two kids (whom I know nothing of) back home, but now needs me to pick them up at the airport. The line goes dead. I show up at the airport and there are 4 pairs of children waiting behind the glass to be claimed, conveniently arranged in pairs of GG, GB, BG, and BB. Each pair is equally likely to be the one I am supposed to pick up.
If my friend had told me "their names are Sarah and--" before the line went dead, I would mentally eliminate BB. Sarah could be any of the four girls. The three remaining options are not equally likely to contain Sarah, because the first one has two girls, and either one could be Sarah. So, the probability of Sarah being the girl in BG is 1/4 and of being the girl in GB is 1/4. The probability of the other child being a girl is also 1/2. It's not so much that BG or GB is eliminated in this case, but that the probability of her being in the GG group is better. But wait a second, I am not trying to find Sarah, I am trying to find the pair of kids that is his pair of kids. Is the probability that a group contains Sarah different from the probability that a group of kids is his? Maybe that's my bad assumption...maybe "at least one is a girl" is some precise formulation I don't understand that means the GG group isn't twice as likely to contain that girl. I'll get back to that. (Possibly her name is "at least once"?).
But what if he instead told me something over the phone that would eliminate the possibility of it being two boys only? What if when he had looked out over the groups of children waiting to be picked up before they blindfolded him and panicked and said "it's not the two boys--"? Is knowing that one of the children is a girl different from knowing they both aren't boys? I think it is, because the latter is a statement about the set of events, and the former is a statement about a single event. When we talk about the pairs of children, the set of events, we are not conveying direct information about the individual events. In this second case he is not conveying information about the individual events and the three remaining choices remain equally likely, 1/3 each.
Looking back to "at least one is a boy"- if we interpret that as a statement about the sets of probabilities, I would more precisely restate Eliezer's question as "Does the set of two children contain at least one boy?". This is why the important part of the story is the word "mathematician". The mathematician is talking about eliminating sets of events when he says "Yes". Ordinary people would just talk about the gender of a child.
You know what's really scary? I saw this out of context, on news.yc/newcomments, and I didn't realize till I came to "pairs of GG, GB, BG, and BB" that it was a fictional setup.
See here for why I think this problem confuses people: http://www.overcomingbias.com/2008/03/mind-probabilit.html