A bag contains $2$ fair tetrahedral dice. The first die has faces labelled $1$, $1$, $2$, and $3$ an

Question

A bag contains $2$ fair tetrahedral dice. The first die has faces labelled $1$, $1$, $2$, and $3$ and the second die has faces labelled $1$, $2$, $3$ and $3$. A die is taken at random from the bag and thrown. The score, $W$ is defined as follows: If the first die is picked and thrown, the score is defined as twice the number which appears on its base. If the second die is picked and thrown, the score is the number which appears on its base. Show that $\mathrm{P}\left( W=2 \right)=\frac{3}{8}$ and find the probability distribution of $W$. [3] Find $\mathrm{E}\left( W \right)$ and $\mathrm{Var}\left( W \right)$. [4]

Accepted Answer

(i) $\mathrm P(W=1)=\dfrac18$, $\mathrm P(W=2)=\dfrac38$, $\mathrm P(W=3)=\dfrac14$, $\mathrm P(W=4)=\dfrac18$, $\mathrm P(W=6)=\dfrac18$ (ii) $\mathrm E(W)=\dfrac{23}{8}$, $\mathrm{Var}(W)=\dfrac{135}{64}$

2022 JPJC P2 Q5

2022 JPJC P2 Q5

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