2023 TMJC P2 Q6

At a particular game stall, players throw darts at a target board to earn points for redeeming prizes.

The target board can be modelled by three concentric circles with radii one unit, three units and five units respectively. A player will get a score of \(50\), \(25\) or \(0\) points if the dart lands on the respective regions labelled \(50\), \(25\) or \(0\) as shown in the diagram above. The probability that a dart lands on any region labelled \(50\), \(25\) or \(0\) is proportional to the area of the region. It can be assumed that the borders between the regions have negligible thickness and the dart will only land on the target board, which has a total area of \(25\pi \) units\(^{2}\).

In a game, a player throws two darts at the target board, one after the other. The first dart will be removed before the next dart is thrown. Let \(S\) be the combined score that the player achieves from the two throws.

1. Show that \(\mathrm{P}\left( S=75 \right)=\frac{16}{625}\).[3]
2. Find the probability distribution of \(S\).[4]
3. Find the expectation and variance of \(S\).[2]