2024 ACJC H1 Q9

There are two bags of balls. Bag \(A\) contains \(2\) red balls and \(1\) black ball. Bag \(B\) contains \(1\) red ball and \(1\) black ball.

1. A bag is selected at random, and the first ball is drawn at random from it.
2. If the first ball is black, it is returned to the selected bag.
3. If the first ball is red, it is placed in the other bag.
4. Regardless of whether the first ball is red or black, the second ball is drawn at random from the other bag.
   

1. Draw a tree diagram to represent this information.[2]
2. Find the probability that both balls are of the same colour.[2]
3. Given that both balls drawn are red, find the probability that the first bag chosen was \(A\).[3]