The difference is in how the information got revealed (whether it was seen or heard is irrelevant). In the case where you saw the girl, it completely specifies

The difference is in how the information got revealed (whether it was seen or heard is irrelevant). In the case where you saw the girl, it completely specifies the gender of the child you saw and unspecifies the gender of the unseen child. (Intuitively: suppose you know the family has four children, and you peek in the window and see that three of them are girls. What's the probability of the fourth being a girl? 1/2.) The case where you are told the family has at least one girl is potentially different... It depends on how you get this information. If you ask the parent, "Do you have at least one girl?" and they say yes, then the probability of the other child being a girl is 1/3. On the other hand, if you ask the parent, "What can you tell me about your children?" and they say "I have at least one girl", then the probability of the other child being a girl is 1/2. The difference is that in the second case, the parent chooses which child to reveal information about, while in the first case, they are forced to make a combined statement about both children. In the second case, the parent completely reveals the gender of the chosen child while unspecifying the gender of the other, just as good as seeing the chosen child in the window. Obviously, most people who describe the boy-girl problem are interested in describing the non-intuitive 1/3 result. But in their rush to reveal the "paradox", many state the problem poorly, leaving open the possibility of the 1/2 case (as Jeff did when he presented the problem). (I added a similar response with more details in the Wikipedia article discussion several months ago: http://en.wikipedia.org/w/index.php?title=Talk:Boy_or_Girl_paradox&oldid=283994010#Responses_to_rebuttals_for_1.2F2_argument http://en.wikipedia.org/w/index.php?title=Talk:Boy_or_Girl_p... I must warn, however, that the concluding paragraph there is incorrect: Jeff's problem is fundamentally undecidable as stated.)