Oh, boy (2)

Yes, we did go wrong, although it's not that easy to see it.  Let's recap - we said that with two children there are four possibilities - BB, BG, GB and GG, and if we know that at least one child is a boy, we can eliminate GG.  So far, so good.  But then we treat the remaining three cases as equally likely - and this is where we go wrong.  If we're going to treat BG and GB as two separate events (boy followed by girl, or girl followed by boy) then we must treat BB as two separate events.  It's easier to see if you give them names - let's say Jim and Fred - so we can have Jim followed by Fred, or Fred followed by Jim.  So we have four possibilities, not three - JF, FJ, BG and GB, of which two satisfy the statement "both are boys" and the probability is therefore 2 in 4, or 1 in 2.  So it's a 50% probability in both cases.