Oh, boy!

Another of those problems which can drive you mad -
(1) A man has two children.  The elder is a boy.  What's the probability that both are boys?
(2) A women has two children.  At least one is a boy.  What is the probability that both are boys?
      OK, if you have two children, there are four possibilities (B for boy, G for girl) - BB, BG, GB or GG. In case (1) we can eliminate GB and GG, so we're left with two possibilities, both equally likely, and the probability that they are both boys is therefore 1 in 2.  In case (2) the only one we can eliminate is GG, so we have three possibilities, and therefore the probability that both are boys is 1 in 3.  But how can knowing the order of birth make a difference?  The chance of any given child being a boy is 50-50 surely? - well close to, anyway.   Where did we go wrong - did we go wrong?  More to come....