To clarify (in case unclear):
"If a green ball or a red ball is drawn on 1st draw,
then the ball is put back in the bag; but not if blue."
The ball that is put back counts as 1st draw;
like, it doesn't mean "start over"; example:
G is drawn; then put back so we have [G R B B] for 2nd draw;
possible to draw G again, so end up with having drawn 2 G's.
And on this:
"What is the probability of ending up with a pair
containing 1 blue and 1 green ball, or 1 blue and 1 red ball?
Really means: what is probability that the 2 balls
drawn are B and G or B and R (order unimportant)
I'm just an imagination of your figment!