Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

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?

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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

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 (GitHub) — ATP and WTA match results, rankings and serve statistics as CSV (CC BY-NC-SA 4.0: non-commercial use, credit the author). The tennis_atp repository was not available when we checked in October 2026; use and cite any copy you find with its date.

Real data for this idea

Primary source: Live Tennis API: academic data access. Students apply here from a school or university email address. Check what the offer includes, then ask only for the matches, players and seasons your question needs. Licence: Check the terms of the academic offer before you use or share the data.
Also useful: Live Tennis API: for teachers. Teachers apply here from a school email address to use the data with a class.

Live Tennis API is listed as a sports-statistics data source only: we have no partnership with it and nothing is paid either way. Its links were added in November 2026, after the October link check. Cite the data and the date you got it. Links checked October 2026; we link to the data and never copy it. More free data sets.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Binomial probabilities for reaching 40–0, 40–15, 40–30; Deuce as an infinite geometric series
AA HL	Good fit	Binomial probabilities for reaching 40–0, 40–15, 40–30; Deuce as an infinite geometric series
AI SL	Good fit	Binomial probabilities for reaching 40–0, 40–15, 40–30; Deuce as an infinite geometric series
AI HL	Good fit	Binomial probabilities for reaching 40–0, 40–15, 40–30; Deuce as an infinite geometric series

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: forgetting that deuce can repeat indefinitely; and treating serving and receiving points identically. Say how each one affects your answer.

Use of mathematics (E)

SL: 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, each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

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).

Extending it for HL

Add an HL technique the question genuinely needs: what an HL probability exploration adds.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete probability exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

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 and comparing with real serve statistics.

Where can I get data for this IA?

Use serve-point win percentages from Jeff Sackmann's ATP data to test the model's game-win predictions. Checked sources and tools: Live Tennis API and Jeff Sackmann's tennis data (GitHub).

Turn this idea into your IA

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