In a game, a fair six-sided die is rolled. If a $1$ or a $2$ is rolled, then the die is rerolled

Question

In a game, a fair six-sided die is rolled. If a $1$ or a $2$ is rolled, then the die is rerolled and the score is the result of the reroll. Otherwise, the score is the number rolled originally. Give the probability distribution function of the scores in the form of a table.[2] Show that the expected score is $\frac{25}{6}$. Explain the meaning of this value in this context.[2] Find the exact variance of the scores.[1]

Accepted Answer

(a) $P(X=1)=P(X=2)=\dfrac1{18}$, and $P(X=x)=\dfrac29$ for $x=3,4,5,6$ (b) $E(X)=\dfrac{25}{6}$ (c) $\operatorname{Var}(X)=\dfrac{73}{36}$

2025 RVHS P2 Q7

2025 RVHS P2 Q7

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