18. (12 points) In an 11-point table tennis match, each point won scores 1 point. When the score reaches 10:10, players alternate serves, and the first player to score 2 more points wins the match. Two students, A and B, play a singles match. Assume that when A serves, A scores with probability 0.5; when B serves, A scores with probability 0.4. The results of each point are independent. After a certain match reaches 10:10 with A serving first, the two players play $X$ more points before the match ends. (1) Find $P ( X = 2 )$; (2) Find the probability of the event ``$X = 4$ and A wins''.
Solution:
18. (12 points)\\
In an 11-point table tennis match, each point won scores 1 point. When the score reaches 10:10, players alternate serves, and the first player to score 2 more points wins the match. Two students, A and B, play a singles match. Assume that when A serves, A scores with probability 0.5; when B serves, A scores with probability 0.4. The results of each point are independent. After a certain match reaches 10:10 with A serving first, the two players play $X$ more points before the match ends.\\
(1) Find $P ( X = 2 )$;\\
(2) Find the probability of the event ``$X = 4$ and A wins''.\\