It is only 1/2 if you fix the order…
We did this a couple of years ago, but at that point it involved children… not buying them obviously, but in that question the answer depended on how the question was asked/answered…
If for example in this question the wife had said something which fixed the sex of one of the dogs, by saying the biggest was a boy, or something like that, then the probability of that dog being a boy is 1. At this point the people saying the answer is 1/2 would be correct.
However, the question is specifically worded to avoid that answer, it is a probability trick often used to demonstrate how sneaky probability can be. So in this case the wife’s answer does not tell you anything specific about either dog, just about the pair of dogs in general. All you know is that one dog is a boy. So is it the biggest dog? Or is it the smallest?
Without the wife’s information the chances that both dogs are boys would be 1/4. Does everyone agree with that?
So, starting from there and then taking the wife’s information into account, it tells us that out of the 4 options, we can ignore the one which has no boys. So, how many options are now left?
Out of those that are left, how many of those have two boys?