Rain or Shine

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

Squash

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

Snooker Frames

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

Snooker

Stage: 5 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 $p = 0.5$?