Ask yourself: of the families that have two children, what are the odds that there is a boy and a girl?
I would say there are four equally likely cases:
BB
BG
GB
GG
and two of them are favorable, so the odds are 50%
You would say,
When working probability Girl-Boy and Boy-Girl is the same thing.
so there are 3 cases - two boys, two girls, and one each.
Three cases, and one of them is favorable, so the odds are 1/3.
Are you still comfortable with your reasoning?