"Under your inference, the man wouldnt have mentioned anything unless he had at least one male child. (in which case you can say the GG scenario is gone, but GB BG and BB are equally probable)"
No. I'm just assuming he has two children, and randomly mentions something about one of them. The GG scenario is only eliminated after he makes his statement, because we then know he has at least one boy.
Put it this way. before he says it, we have
GG, GB, BG, BB
after he says "I have a child that is [MALE OR FEMALE]" we have (where the capital letter is the child whose sex has been mentioned, and the lowercase letter is the other child):
Gg, gG, Gb, gB, Bg, bG, Bb, bB
So if he has said the sex is male, then we have four combinations left:
gB, Bg, Bb, bB.
Understand that we go to more scenarios (8) based on which child is mentioned, before we go to fewer. Actually the order of the children is something you can and should ignore, however as you are holding on to it, I show it this way....
I'm afraid this is wrong - you shouldn't distinguish between Bb and bB. In this problem, they are not different states, so counting them messes up your probability calculation.
No. I'm just assuming he has two children, and randomly mentions something about one of them. The GG scenario is only eliminated after he makes his statement, because we then know he has at least one boy.