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]