Challenge Level

In a particular snooker tournament, two players play "frames" against each other and the first to win 8 frames wins the match. This is known as a "best of 15" match. Assume that the probability of player A winning each frame is $p$, regardless of who starts. If A does not win the frame then his opponent does (there are no draws).

Suppose $p = 0.4$, then what is the probability that A wins the match? Now suppose that A is very slightly better than his opponent, say $p = 0.55$, what is the probability now of A winning the match? What is the probability of A winning the match when $p = 0.5$?