2025 DHS P2 Q8

The bag contains \(6\) red discs, \(6\) green discs, \(6\) yellow discs and \(6\) blue discs. The discs are indistinguishable aside from their colour. \(4\) discs are taken one after another from the bag without replacement. The random variable \(X\) is the number of different colours obtained among the \(4\) discs.

1. Show that \(\mathrm{P}(X=2) = \frac{465}{1771}\).[3]
2. Hence determine the probability distribution of \(X\).[3]

Joe plays a game that comprises \(8\) rounds of drawing \(4\) discs from the bag. For each round, he wins \$10 if he obtains at most \(2\) discs of different colours, otherwise he loses \$5. After each round, all the discs are returned to the bag before the next round.

3. Find the probability that Joe wins some money at the end of the game.[2]
4. Explain, with working, whether it is worthwhile for Joe to play this game to win money.[2]