Timothy runs a game stall at a funfair. In a particular card game, there is a deck of \(\left( n+4 \

Question

Timothy runs a game stall at a funfair. In a particular card game, there is a deck of $\left( n+4 \right)$ cards consisting of four cards numbered $5$ and $n$ cards numbered $3$, where $n\ge 2$. To play a game, a player draws two cards at random and without replacement from the deck of cards. The random variable $M$ represents a player’s score, which is the product of the scores of the two cards drawn in one game.

Determine the probability distribution of a player’s score in one game.[3]

For the card game, the payout of one game in dollars, $W$, is such that $W=\left| M-15 \right|$.

Given that $\mathrm{E}\left( W \right)=\frac{100}{23}$, find the value of $n$.[3]
Using the value of $n$ found in (ii), find the probability that the mean payout of $40$ independent games is less than $\$4.50$.[3]

Accepted Answer

(i) $\mathrm P(M=9)=\dfrac{n(n-1)}{(n+4)(n+3)}$, $\mathrm P(M=15)=\dfrac{8n}{(n+4)(n+3)}$, $\mathrm P(M=25)=\dfrac{12}{(n+4)(n+3)}$ (ii) $n=20$ (iii) $\mathrm P(\overline W<4.50)=0.633$ to $3$ significant figures

2024 SAJC P2 Q6

2024 SAJC P2 Q6

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