The sex of the second child is not affected at all by the sex of the first child; knowing that one of the children is a girl is irrelevant in detemining the sex of the other.
Trust me, I'm a Doctor (of biology)
I agree... the selection process has no memory. It doesn't remember the sex of the first element when choosing the second. This is like throwing a coin. Each throw is independent, since the coin has no memory. It is not like opening two doors where
behind one is a prize. In that case after opening the first one, the second one will hold the prize because there's "memory" involved.
You could easily transform this given problem into the coin scenario. But you can't into a door problem because the mother can't say: OK first one was a boy/girl therefore the second one is of the opposite sex
Also, Mr. Biologist, the odds are not 50-50. There are slightly more girls on this world