On reflection, let me withdraw what I said earlier. I think I see where the issue is.
The poin of this problem, and the point of the exact phrasing here, is that the two children in question are undifferentiated. As soon as you differentiate the two, the problem changes. The disconnect here is that you are differentiating them, into “the child the parents tell you about” and “the child the parents don’t tell you about.” That results in exactly the same problem as if you had differentiated the two into “the older” and “the younger.”
So, going back:
Suppose the problem were stated this way: “There is a family with two children. You have been told the eldest child in this family is a daughter. What are the odds they also have a son, assuming the biological odds of having a male or female child are equal?” In this case, the correct answer is 1/2.
Likewise, suppose the problem were stated this way: “There is a family with two children. The parents pick a child, and tell you it is a daughter. What are the odds they also have a son, assuming the biological odds of having a male or female child are equal?” That’s your re-interpretation of the problem. In this case, the correct answer is also 1/2.
But the real problem isn’t stated this way. It’s stated: “There is a family with two children. You have been told this family has a daughter. What are the odds they also have a son, assuming the biological odds of having a male or female child are equal?” And the answer is 2/3.
I think we’re sort of in agreement now, or at least you see where I was coming from. I think part of it comes from how you’re taught to listen to these things. The teacher who taught me how to handle statistics basically said "when a problem states ‘you are told’, you have to take that into consideration. If something is simply stated (ie, had it been phrased ‘the family has a daughter’, with the ‘you are told’ omitted), then you do not factor in that probability. Her reasoning was that whenever you are told something rather than it being independently revealed or discovered, there is always a decision there, whether implied or stated.
Not quite. Always assuming honesty, he had no choice but to answer yes in that instance. However, had the question been more open, with “What is the gender of exactly one of your children?”, he would have had to make the decision to state the gender of one or the other of his children.
If the problem meant us to tag daughterhood on a particular child — whether that be the youngest, the oldest, the tallest, the heaviest, the first in the alphabet, the one being thought of right now by somebody, or whatever — then it’s crucial this fact be in the problem text somewhere because, as shown in this thread, it changes the answer entirely.
The sentence, “You have been told this family has a daughter”, maybe does suggest at first blush that a particular child has been selected (the one in the mind of person telling us this), and is being identified as female. It sort of suggests that the person has seen one or both of the children, or been told about them by another reliable source, and is now passing some of that information along to us. Or something like that.
But, we could equally well be reading the result of a database query, telling us merely that this family has exactly two children and at least one daughter, and no other facts. Or perhaps our informer once drove by the family’s house, saw an Easy-Bake™ Oven on the porch, and — knowing this county’s strict laws against boys playing with Easy-Bake Ovens — concluded pretty soundly that the family must have a girl or two.
We don’t know what extra information might lie in the head of our informer, beyond what we’re given. Knowledge that a particular child is female would count as extra information. The sentence, “You have been told this family has a daughter”, hints that a particular child is being thought of and talked about — but really it is saying only, “This family has at least one daughter.”