Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won

Question

Players $A$ and $B$ compete in a racquet match consisting of at most $3$ sets. Each set is won by either Player $A$ or $B$, and the match is won by the first person to win two sets. Player $A$ has a probability of $\frac{2}{3}$ of winning each of the first two sets. If the match goes into the third set, Player $A$ has a probability of $p$ of winning this set. With the aid of a tree diagram, find in terms of $p$, the probability that player $A$ will win the match. Deduce the range of values of the probability that player $A$ wins the match. Show that the value of $p$ in order for the match to be fair is $\frac{1}{8}$. Given that Player $A$ wins the match, find, in terms of $p$, the probability that he won the $2^{\text{nd}}$ set.

Accepted Answer

(i) $\frac49(1+p)$ (ii) $\frac49\le P(A\text{ wins})\le\frac89$ (iii) $p=\frac18$ Given $A$ wins: $\frac{p+2}{2(p+1)}$

2011 NYJC Prelim

2011 NYJC Prelim

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