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

(b)min = -0.06, max = -0.31 (c)$P(\bar{X}\geq 0)=0.397$ (d)0.188 (e)$\frac{3}{11}$ or $\frac{4}{11}$

2025 HCI P2 Q9

2025 HCI P2 Q9