Of those three outcomes, only one has two boys, so the answer of 1/3 is indeed justified.

But some people still think the answer should be 1/2. Their reasoning is "If one child is a boy, then there are two equiprobable outcomes for the other child, so the probability that the other child is a boy, and thus that there are two boys, is 1/2."

When two methods of reasoning give two different answers, we have a paradox. Here are three responses to a paradox:

The very fundamentals of mathematics must be incomplete, and this problem reveals it!!! I'm right, and anyone who disagrees with me is an idiot!!! I have the right answer for one interpretation of the problem, and you have the right answer for a different interpretation of the problem.

If you're Bertrand Russell or Georg Cantor, you might very well uncover a fundamental flaw in mathematics; for the rest of us, I recommend Response 3. When I believe the answer is 1/3, and I hear someone say the answer is 1/2, my response is not "You're wrong!", rather it is "How interesting! You must have a different interpretation of the problem; I should try to discover what your interpretation is, and why your answer is correct for your interpretation." The first step is to be more precise in my wording of the experiment:

Child Experiment 2a. Mr. Smith is chosen at random from families with two children. He is asked if at least one of his children is a boy. He replies "yes."

The next step is to envision another possible interpretation of the experiment:

Child Experiment 2b. Mr. Smith is chosen at random from families with two children. He is observed at a time when he is accompanied by one of his children, chosen at random. The child is observed to be a boy.