A new game has been designed for a particular casino using two fair die. In each round of the game,

Question

A new game has been designed for a particular casino using two fair die. In each round of the game, a player places a bet of $\$2$ before proceeding to roll the two die. The player’s score is the sum of the results from both die. For the scores in the following table, the player keeps his bet and receives a payout as indicated.

Score	Payout
9 or 10	$\$1$
2 or 4	$\$5$
11	$\$8$

For any other scores, the player loses his bet.

Let $X$ be the random variable denoting the winnings of the casino from each round of the game.

Show that $\mathrm{E}\left( X \right)=\frac{1}{12}$ and find $\mathrm{Var}\left( X \right)$.[4]
$\bar{X}$ is the mean winnings of the casino from $n$ rounds of this game. Find $\mathrm{P}\left( \bar{X}>0 \right)$ when $n=30$ and $n=50 000$.
Make a comparison of these probabilities and comment in context of the question.[3]

Accepted Answer

(i) $E(X)=\frac1{12}$, $\operatorname{Var}(X)=\frac{1307}{144}$; (ii) for $n=30$, $P(\bar X>0)=0.560$; for $n=50000$, $P(\bar X>0)=1.00$

2017 DHS P2 Q5

2017 DHS P2 Q5

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Solution:

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Score	Payout
9 or 10	\(\$1\)
2 or 4	\(\$5\)
11	\(\$8\)