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# Snooker Frames

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Age 16 to 18

Challenge Level

A single game of snooker is called a frame. In the first round of a snooker tournament, the matches are played over eleven frames (so the first player to win six frames wins the match). In the later rounds matches are played over 15 frames. Assume that each player has steady nerves and his chance of winning any frame (irrespective of who starts) is constant.

In the problem Snooker you were asked to find the probability that a player wins a match over fifteen frames, given that his chance of winning any frame is $0.4$. You should now find the probability that this player wins a match over eleven frames.

It is believed that the weaker players have a better chance of winning the matches over eleven frames than they do over fifteen frames. Do your results confirm this or not?

Does this surprise you, or not? Why?

Numerical
investigation: Is it generally the case that more frames
lead to a reduced chance of a weaker player winning? Devise a
spreadsheet which computes the chance of a weaker player winning a
match in which the first player to $1, 2, 3, 4, \dots, 17, 18$
frames wins. The world snooker championship final is taken over the
best of $35$ frames. In order to have at least a $10\%$ chance of
winning such a final, what probability of winning each frame would
you need to have? Plot $\log(n)$ against $P(win)$.

Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

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?

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?