2018 PJC P2 Q6

A circular card is divided into \(3\) sectors scoring \(0\), \(1\), \(2\) and having angles \({{135}^{\circ }}\), \({{135}^{\circ }}\), \({{90}^{\circ }}\) respectively. The card is mounted onto a wall at its centre such that it can rotate freely. A pointer is fixed on the wall just above the card, as shown in the diagram

In a game, a participant will spin the card twice, and the random variable \(X\) is the product of the scores of the \(2\) spins.

1. Tabulate the probability distribution of \(X\).[2]
2. Show that \(\mathrm{E}\left( X \right)=\frac{49}{64}\), and find \(\mathrm{Var}\left( X \right)\).[2]

Dave intends to use the above setup in a fundraising carnival. For every game, the participant will be charged \(\$y\) to play, and will be awarded a prize money of \(\$5X\) based on the outcome of their spins. Find the least integer value of \(y\) such that on average, Dave can expect to earn at least \(\$1\) from each participant.[2]