2022 RI P2 Q8
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.
(i)
Show that the probability that a game results in a score of $4$ is $\frac{7}{32}$.
[2]
(ii)
Find the expectation and variance of the score.
[5]
(iii)
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]
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