2024 TMJC P2 Q6

At a carnival, a game stall offers a game of throwing darts. The darts will land either in the bull’s eye, inner ring or outer ring. There is a probability of \(p\), \(2p\) and \(5p\) of landing in the bull’s eye, inner ring and outer ring respectively.

A player gets five points if the dart lands in the bull’s eye, three points if the dart lands in the inner ring and one point if the dart lands in the outer ring. A round consists of two throws. Let \(X\) be the total number of points for one round of two throws.

1. Find the value of \(p\) and show that \(P\left( X=6 \right)=\frac{7}{32}\). Obtain the probability distribution of \(X\).[5]
2. Find \(\mathrm{E}\left( X \right)\).[2]
3. A player wins \(\$1\) for every two points earned. If a player pays \(\$k\) to play one round of two throws, find the value of \(k\) for the game to be fair.[2]