Look at it like this: there are two children, one who opens the door, and one who doesn't. Before she opens the door, the possibilities are:
Door Opener / Non Door opener
Boy / Girl
Boy / Boy
Girl / Girl
Girl / Boy
Once she opens the door, the first two possibilities are eliminated, leaving only the options that the shy child is either a Boy or a girl. In other words, the sex of the one child has nothing to do with the sex of the other.
Prove me wrong.
EDIT:
Even better, look at it like this:
FAMILIES WITH TWO CHILDREN:
25% have two boys - A girl will answer the door 0% of the time
50% have a boy and a girl - A girl will answer 50% of the time
25% have two girls - a girl will answer 100% of the time
If we take the product of the two percentages in each row, we get some more numbers to cloud the issue:
PROBABLILITY THAT A GIRL WILL ANSWER...
...and the family has two boys - 0%
...and there's a boy and a girl - 25%
...and the family has two girls - 25%
So, a girl will answer (0% + 25% + 25%) = 50% of the time.
Exactly half of those responses will come from families with two girls, and exactly half will come from families with a boy and a girl.
Or I could be completely off base - let me know.