the paradoxical result that when we select a family the probability is ⅓ but when we select the boy the probability is ½. Hence the title "boy-girl paradox".
Well it only feels like a paradox, like many paradox it isn't actually a paradox.
For example if you and me were in my kitchen (and I had two children) and my children were playing outside but you couldn't see them, and you asked if at least one of my children was a boy (odd question but lets role with it) and I said yes. Then in this scenario the chance of both being boys would be 1/3. Note the important thing here is that I didn't select a random true statement, if both my children were girls I would have answered no to your question. To explain this I like to think of cards A, B, C and D. A = BB, B = BG, C = GB, D = GG. Your question has eliminated any chance of D. This leaves A, B and C. The chance of any one of those cards being the right answer is 1/3.
However in the same scenario, you and me are sitting in my kitchen, children playing outside. And you can see one of the children (and it's a boy). Then in this scenario the chance of the other being a boy is 1/2. This feels like a paradox, what is the difference between seeing one is a boy and knowing one is a boy? The difference in this case is that looking at a boy is testing one boy not both boys. For example with the card example of A, B, C and D. We obviously remove D. But as we have seen the boy then either B or C is also removed, depending on if you saw my older or younger child (note it doesn't matter if you know which is older or younger). So we are left choosing between A and (B or C) which gives the answer of 1/2.
The results are counter intuitive and confusing. We simply aren't well equipped for this to make sense. Some people simply can't accept the answer, which I find interesting. You (Fooloso4) even if intuitively you find it hard to believe do defer to the wiki article, which personally I find quite refreshing
Unknown means unknown.