In each round of a treasure hunt game, a player randomly selects a spot from a large number of prede

Question

In each round of a treasure hunt game, a player randomly selects a spot from a large number of predefined treasure locations on an island. Each spot uncovers one outcome, and a score $x$ is awarded based on the outcome. The table below shows all the possible outcomes in a round and the corresponding scores. Each round is independent and the game resets after every round, so that the probabilities remain unchanged. OutcomeCursed trapSmall trapMystery boxGold chest Score, $x$$-3$$-0.3$$p$$5$ $P(X=x)$$0.2$$p$$q$$0.1$ Show that $E(X)=-0.1+0.4p-p^2$.[2] Hence find the maximum and minimum possible values of $E(X)$.[2] The treasure hunt game is played for $30$ rounds. If $p=0.4$, find the probability that the player's mean score exceeds $0$.[3] The treasure hunt game is played for $10$ rounds. Given that the probability of finding more than $3$ Small traps in these $10$ rounds is $10\%$, find the value of $p$.[2] Over a long period of time, it is observed that the number of Small traps found in $10$ rounds of the game follows a bimodal distribution, with one of the modes being $3$. Find the two exact possible values of $p$, showing your working clearly.[3]

Accepted Answer

(a) $E(X)=-0.1+0.4p-p^2$ (b) minimum $-0.31$, maximum $-0.06$ (c) $0.397$ (d) $0.188$ (e) $p=\dfrac3{11}$ or $p=\dfrac4{11}$

Outcome	Cursed trap	Small trap	Mystery box	Gold chest
Score, \(x\)	\(-3\)	\(-0.3\)	\(p\)	\(5\)
\(P(X=x)\)	\(0.2\)	\(p\)	\(q\)	\(0.1\)

2025 HCI P2 Q9

2025 HCI P2 Q9

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