How much does a small edge on each point matter in tennis?

AA SLAA HLAI SLAI HL Solid Also in: Sport, Games & puzzles

Research question

If a player wins each point on serve with probability p, what is the probability that they win a game, a set and a match, and how does tennis scoring amplify a small advantage?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Winning 55% of points can mean winning far more than 55% of matches. Working out exactly how the scoring system magnifies a small edge uses binomial probability, geometric series (deuce!) and a model you can check against real match statistics.

The mathematics you'll need

Binomial probabilities for reaching 40–0, 40–15, 40–30
Deuce as an infinite geometric series
Building up from games to sets (recursion or a tree)
Graphing P(win game) against p and interpreting the S-shape
Comparing with real serve statistics

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use serve-point win percentages from Jeff Sackmann's ATP data to test the model's game-win predictions.

Jeff Sackmann's tennis data — ATP match results, rankings and serve statistics as CSV (CC BY-NC-SA 4.0; a WTA repository exists too).

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

State the independence assumption and why it is questionable.
Derive P(win game) step by step, including deuce.
Extend to sets (tie-break simplified if needed).
Compare predicted and actual service-game win rates.
Reflect on momentum and pressure points.

Pitfalls that cost marks

Forgetting that deuce can repeat indefinitely.
Treating serving and receiving points identically.
Jumping to a formula from a website without deriving it.

Showing personal engagement

Use your own serve statistics if you play.
Compare with badminton or table-tennis scoring.
Design a scoring system that reduces the amplification and justify it.

See Criterion C: personal engagement for what examiners look for.

Taking it further

HL: set up a Markov chain for a game and verify the result with matrix powers (AI HL) or prove the deuce formula rigorously (AA HL).

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).