I think the "random" part is important. If you just said, "a lady has two children, one is a son. What are the odds that the other is a son", I think it's more ambiguous. If you went out and found a lady that had a son, then the odds that the other is a son would be, I think, 50%. But if you found a random person, determined that she had one son, and asked what the other might be, that's different.
I think. 
No. If you only sought out a lady who had a son and one other child, and you know nothing else about the lady, then there is no difference.
However, if a lady has one son, and she is pregnant, the probability that the child-to-be is a boy is 50%. It seems, superficially, that the situations are identical. After all, the lady will eventually have one boy, and one other child. But giving birth shouldn't change the probability that the other child is a boy.
I'll pose this as a new problem
. Explain the difference that accounts for the difference in probability.
