Archer and Betty took part in a competition comprising of at most $3$ games. Each game is either w

Question

Archer and Betty took part in a competition comprising of at most $3$ games. Each game is either won by Archer or Betty. The first person who win $2$ games wins the competition. The probability of Archer winning the first game is $0.25$. The probability of him winning any subsequent games is $p$ and is independent of any previous games.

Draw a probability tree diagram to represent the above information.[1]
Find, in terms of $p$, the probability that Archer will win the competition.[2]

For the rest of the question, use $p=0.5$.

Find the probability that Betty won the second game, given that she won the competition.[3]

The number of games won by Archer in a competition is denoted by $W$. Using $p=0.5$,

determine the probability distribution of $W$,[2]
find $\mathrm{Var}\left( W \right)$.[2]

Accepted Answer

(ii) $0.5p\left( 1+p \right)$ (iii) $0.625$ (iv) $\frac{3}{8}$, $\frac{1}{4}$, $\frac{3}{8}$ (v) $0.75$

2022 ACJC P2 Q7

2022 ACJC P2 Q7

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