Bag $A$ contains 4 balls numbered 2, 4, 5, 8. Bag $B$ contains 5 balls numbered 1, 3, 6, 8, 8. Bag $C$ contains 7 balls numbered 2, 7, 8, 8, 8, 8, 9. One ball is selected at random from each bag.
a) Find the probability that exactly two of the selected balls have the same number.
b) Given that exactly two of the selected balls have the same number, find the probability that they are both numbered 2.
c) Event $X$ is ‘exactly two of the selected balls have the same number’. Event $Y$ is ‘the ball selected from bag $A$ has number 2’. Showing your working, determine whether events $X$ and $Y$ are independent or not.
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