N2008 P2 Q7

A computer game simulates a tennis match 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 simulation uses the following rules.

• The probability that \(A\) wins the first set is 0.6.
• For each set after the first, the conditional probability that \(A\) wins that set, given that \(A\) won the preceding set, is 0.7.
• For each set after the first, the conditional probability that \(B\) wins that set, given that \(B\) won the preceding set, is 0.8.

Calculate the probability that

1. \(A\) wins the second set,[2]
2. \(A\) wins the match,[3]
3. \(B\) won the first set, given that \(A\) wins the match.[3]