2017 HCI P2 Q6

A biased tetrahedral (\(4\)-sided) die has its faces numbered ‘\(-1\)’, ‘\(0\)’, ‘\(2\)’ and ‘\(3\)’. It is thrown onto a table and the random variable \(X\) denotes the number on the face in contact with the table. The probability distribution of \(X\) is as shown.

1. The random variable \(Y\) is defined by \({{X}_{1}}+{{X}_{2}}\), where \({{X}_{1}}\) and \({{X}_{2}}\) are \(2\) independent observations of \(X\). Show that \(\mathrm{P}\left( Y=2 \right)=\frac{3}{16}\).[2]
2. In a game, a player pays \(\$2\) to throw two such biased tetrahedral dice simultaneously on a table. For each die, the number on the face in contact with the table is the score of the die. The player receives \(\$16\) if the maximum of the two scores is \(-1\), and receives \(\$3\) if the sum of the two scores is prime. For all other cases, the player receives nothing. Find the player’s expected gain in the game.[4]