2017 HCI P2 Q6

Timothy Gan

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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.

(i)

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 $\text{P}\left( Y=2 \right)=\frac{3}{16}$.

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(ii)

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.

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