Tim plays a game with $3$ outcomes having the score of $2$, $5$ and $6$. The corresponding p

Question

Tim plays a game with $3$ outcomes having the score of $2$, $5$ and $6$. The corresponding probabilities are $k$, $\frac{2}{7}$ and $\frac{4}{7}$ respectively, where $k$ is a constant. State the value of $k$.[1] Tim pays $\$m$ in total to play the game twice. The outcome of each game is independent. He receives an amount corresponding to twice the absolute difference of the two scores from each game. If Tim is expected make a profit, determine the range of values of $m$.[3] For $m=2.5$, explain why Tim may not profit for every $2$ such games that he plays.[1]

Accepted Answer

(a) $k=\frac17$. (b) $\displaystyle 0\le m<\frac{120}{49}$. (c) For $m=2.5$, payouts $W=0$ or $W=2$ produce a loss, so profit is not guaranteed; moreover $\mathrm E(W)=120/49<2.5$.

2023 DHS P2 Q5

2023 DHS P2 Q5

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