2023 NYJC P2 Q7

A biased four-sided die with faces numbered \({1}\), \({2}\), \({3}\) and \({4}\) is used in a game. The random variable \({X}\) represents the number shown on the face which lands on the table.

The probability of landing on a \({4}\) is \({0.05}\) and the probability of landing on a \({3}\) is \({0.11}\). There is equal probability of landing on either a \({1}\) or a \({2}\).

In a game, players are to throw the die twice and the two independent observations \({{X}_{1}}\) and \({{X}_{2}}\) produce the score \({Y}\) defined as

\(Y=\left\{ \begin{matrix} 4\,\,\,\,\,\,\,\,\mathrm{if}\,\,\,\,\,\,\,\,{{X}_{1}}=4, \\ {{X}_{1}}+{{X}_{2}}\,\,\,\,\,\,\,\,\mathrm{if}\,\,\,\,\,\,\,\,{{X}_{1}}\ne 4. \end{matrix} \right.\)

1. Find \(\mathrm{P}\left( Y=y \right)\) for all possible values of \(y\).[4]
2. Find \(\mathrm{E}\left( Y \right)\) and \(\mathrm{Var}\left( Y \right)\).[2]

Players pay \({\$5}\) for a game. If the score \(Y\) is \({5}\) or \({6}\), the player wins \({\$20}\). If the score \(Y\) is \(7\), the player wins \({\$50}\), otherwise the player wins no money. In a typical day, a player plays \(40\) games.

3. Find the expected loss of the player in a typical day.[2]