A game is played by throwing a fair coin and two fair four-sided dice. The dice are coloured red or

Question

A game is played by throwing a fair coin and two fair four-sided dice. The dice are coloured red or blue, and have faces numbered from $1$ to $4$. If the coin shows a head, then the score is the number shown on the red die. Otherwise, the score is the sum of the numbers shown on the two dice. Show that the probability that a game results in a score of $4$ is $\frac{7}{32}$.[2] Find the expectation and variance of the score.[5] The game is played $35$ times. Estimate the probability that the average score of the $35$ games is at least $4$, given that the first and second games result in a score of $3$ and $4$ respectively.[3]

Accepted Answer

(i) $P(S=4) = \frac{7}{32}$ (ii) $E(S) = \frac{15}{4}$, $\text{Var}(S) = \frac{55}{16}$ (iii) $0.193$ ($3$ s.f.)

2022 RI P2 Q8

2022 RI P2 Q8

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