N2008 P2 Q7 [Modified]

A computer game simulates a tennis match between two players, \({A}\) and \({B}\). Each set is won by either \({A}\) or \({B}\). The simulations use 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 \({0.8}\).

1. In one variation of the game, the match consists of at most three sets and the match is won by the first player to win two sets.
   Calculate the probability that
   1. \({A}\) wins the second set[2]
   2. \({A}\) wins the match[3]
   3. \({B}\) wins the first set, given that \({A}\) wins the match[3]
2. In another variation of the game, there is no limit to the number of sets, and a match is won only when a player wins two consecutive sets. Show that the probability that \({A}\) wins the match is \({\frac{119}{235}}\).[3]