From a bag containing $5$ discs numbered $1$ to $5$ respectively, two discs are drawn at random and

Question

From a bag containing $5$ discs numbered $1$ to $5$ respectively, two discs are drawn at random and without replacement. Let $X$ represent the absolute difference between the numbers on the two discs. Show that $\mathrm{P}\left( X=1 \right)=0.4$ and find the probability distribution of $X$. Find $\mathrm{E}\left( X \right)$ and $\mathrm{Var}\left( X \right)$. In a fund-raising game, two such bags are placed on a table. A player chooses two discs at random and without replacement from each bag. If the value of $X$ is the same for both bags, the player wins $\$X$ ,otherwise the player wins nothing. Find the probability that the player wins a prize. The organisers wish to make an average profit of $\$0.10$ per game. How much should they charge per game?

Accepted Answer

(i) distribution $P(X=1)=0.4$, $P(X=2)=0.3$, $P(X=3)=0.2$, $P(X=4)=0.1$; (ii) $E(X)=2$, $\operatorname{Var}(X)=1$; (iii) $0.3$; (iv) charge $$0.60$ per game

2002 ACJC P2 Q29

2002 ACJC P2 Q29

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