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### Number and algebra

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### Advanced mathematics

# Any Win for Tennis?

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Age 16 to 18

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This is an interesting and complicated scenario which will require clear probabilistic thinking. There will be various different ways in which the first part of the problem can be attempted, and it would be informative to compare and contrast these. It is quite a difficult problem and could be set to keen students individually or discussed collectively. It would be good as revision of a module on probability or to motivate the advanced study of probability.

The main thing to realise is that this modelling situation is more complicated than it might first appear: with a simple counting approach to probability, there is the risk of incorrectly computing an answer without realising it, so require that all discussion is clearly reasoned. Be prepared to negotiate these probabilistic discussions, as they can be quite difficult for learners to
understand.

The first phase is to encourage learners to get to grips with the problem, and the mathematical description of tennis should stimulate some fun and discussion! Hopefully the discussion and learning can be navigated so that learners realise:

- There are different sorts of winning sequences, such as WWWLW, WWLLW.
- Each winning sequence ends with a W.
- There are arbitrarily long winning sequences WWWLLL(WL)....(WL)WW.
- Not all winning sequences are of the same probability.

Once this framework is understood, learners can attempt to work out ways to represent the situation and work out the probability.

The second part of the problem will require learners to work out some representation using technology, either with a spreadsheet or a piece of computer code. This is a valuable learning exercise, and the first step could be to verify numerically the answer to the first part of the question. For this second part, the probabilities will be numerical, so answers will necessarily be approximate;
learners might need to be encouraged that this is OK!

What is the shortest possible game?

How many different sorts of games result in two losses and four wins?

What is the probability of a game hitting deuce? (i.e. 3 points each)

How can a game be represented?

The second part of this question offers plenty of open extension.

You could start with either of these simpler, related problems:

1) The winner is the first to three points.

2) The winner is the first to be two points clear of his or her opponent.