2012 JJC P2 Q7

A tennis match is played between two players, \(A\) and \(B\). The match consists of at most three sets. Each set is won by either \(A\) or \(B\), and the match is won by the first player to win two sets. The probability that \(A\) wins the first set is \(p\).

For each set after the first, the conditional probability that \(A\) wins the set, given that \(A\) won the preceding set, is \(0.7\).

For each set after the first, the conditional probability that \(B\) wins the set, given that \(B\) won the preceding set, is \(0.6\).

Construct a probability tree showing this information.[2]

1. When a match is decided by playing all three sets, the probability of \(A\) winning is the same as that of \(B\) winning. Show that the value of \(p\) is \(\frac{8}{11}\).[2]
2. Find the probability that \(B\) wins the first set, given that \(A\) wins the match.[3]