Is this basically to do with the fact that you know how likely it is to see a boy given any of the combinations? So there's a 33% chance that there are two boys and you see a boy, a 33% chance that there's a boy and girl and you see a boy, and a 33% chance that there's a boy and a girl and you see a girl. Given that you've seen the boy you've eliminated one of the outcomes so the other two outcomes are equally likely?

I'm confused by your comment but I'll try to explain more.

Given one of the children is a boy. The only combinations possible are: GB, BG, & BB. Where G is Girl and B is Boy. The question is about the probability of the other child being a Girl.

When we eliminate one of the B from each combination, we get:

GB, BG, BB -> G, G, B

We can see that 2/3 are G and 1/3 are B.

Does this help?

But are those ordered pairs or sets of two items each?

I think of them as ordered pairs, but would it change anything if they were sets? Order doesn't matter in this question.

Order doesn't matter but it helps make it more clear as why I included the same combination twice.