A game between two players uses a fair 4-sided dice with faces numbered $1$, $2$, $3$ and $4$. In a turn, the dice is thrown repeatedly, with no more than three throws allowed. As soon as a $4$ appears, the turn ends and no further throws are taken in that turn. Any player who throws a $4$ in a turn earns $1$ point.
(a)[2]
Show that the probability that a player gets a $4$ in one turn is $\frac{37}{64}$.
(b)[1]
Xeno and Yao take part in this game. Find the probability that neither Xeno nor Yao scores any points in their first two turns.
(c)[3]
Xeno and Yao each take three turns. Find the probability that Xeno scores $2$ more points than Yao.
Worked solution & mark scheme
This 6-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Valid method for the probability of at least one 4 in 3 throws, e.g. $1-\left(\frac{3}{4}\right)^3$ or an equivalent form.” …