2024 YIJC P2 Q9

A box contains two balls numbered \(2\) and \(k\) balls numbered \(3\), where \(k\ge 3\). In a game, three balls are drawn from the box at random and the score is obtained by multiplying together the numbers indicated on the balls that were drawn. Let the random variable \(X\) denote the score obtained.

1. Show that \(\mathrm{E}\left( X \right)=\frac{9\left( 3{{k}^{2}}+3k+2 \right)}{\left( k+1 \right)\left( k+2 \right)}\).[4]
2. Given that \(\mathrm{E}\left( X \right)=\frac{144}{7}\), find the value of \(k\).[1]
3. Assuming that each game is independent of one another, find the probability that the average score of \(30\) games is more than \(21\).[3]