The model answer for (a) and (b) is 0.16308 and 0.4868 respectively. But I cannot derive the answer for question (b) onwards. I would appreciate your help

6. (Total: 14 points) Alice plays Betty in the final of a table-tennis tournaments. The match consists of five independent games. Whoever wins three games first becomes the champion. In each game, the probability that Alice wins is 0.3 and the probability that Betty wins is 0.7. (a) (2 points) Find the probability that Alice becomes the champion. (b) (2 points) Given that Alice becomes the champion, what is the probability that Betty has won exactly two games? (c) The rules of the match are now revised as follows: Whoever wins three games before the other wins two becomes the champion. If both win two games then they have a deuce, and they will continue to play and the one who first wins two more games than the other becomes the champion. (i) (2 points) Show that the probability that a deuce will ever occur is 0.2646. (ii) (2 points) Given that a deuce has occurred, what is the probability that Alice becomes the champion? (iii) (2 points) What is the probability that Alice becomes the champion without any deuce? (iv) (2 points) Using the results of (i), (ii) and (iii), find the probability that Alice becomes the champion. (v) (2 points) Given that Alice becomes the champion, what is the probability that Betty has won exactly two games