IB Maths IA idea · Sport and games

From rally to match: how small edges become big wins

Win 55% of the rallies and you would expect to win about 55% of the games. You will not. You are going to find out how far off that guess is.

Sport and games
SL good
HL ideal
Maths AA Maths AI Mixed data
Syllabus: AA 1.3, AA 1.8, AA 1.9, AA 4.5, AA 4.8, AA 5.6

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

A match is built from games, and a game is built from rallies. If you win each rally with the same chance p, the chance of winning a game to 11 is a sum of binomial-type terms, plus a tail for the stage where both players reach 10 points and must win by two. That tail is a geometric series, and it sums to a neat expression. Then a match is built from games the same way. The derivative at p = 0.5 tells you how strongly each stage amplifies a small edge. The formulas are the easy part. Say whether rallies are really independent with the same p, test the model against real scores or a simulation, and compare sports with different scoring rules.

The maths you will use

Binomial and negative-binomial counting. Count the ways to win 11 points before the opponent reaches 10: $\binom{10+k}{k}p^{11}q^{k}$ for $k = 0,\ldots,9$ .
Geometric series. Sum the chances of winning from deuce: $p^{2}\sum (2pq)^{n}$ , giving $\frac{p^{2}}{p^{2}+q^{2}}$ .
Composition of functions. Build the match chance as $M(G(p))$ , with $M(g)=g^{2}(3-2g)$ for best of three.
Differentiation. Find $G'(0.5)$ and $M'(0.5)$ as amplification factors; use the chain rule for $M(G(p))$ .
Simulation. Simulate games with a random number generator and compare proportions with the formula, with a stated error.

Key equations

G(p) = \sum_{k=0}^{9}\binom{10+k}{k}p^{11}q^{k} + \binom{20}{10}p^{10}q^{10}\,\frac{p^{2}}{p^{2}+q^{2}}

M(g) = g^{2}(3-2g), \qquad \frac{dM}{dp} = M'(G(p))\,G'(p)

The maths in action

Two panels. Each shows S-shaped curves rising steeply through the point (0.5, 0.5), above a dotted diagonal. The curve for 21 points is steeper than the one for 11, and match curves are steeper still. — (a) Chance of winning a game against the chance of winning a rally, for games to 11 and 21 (win by 2). At p = 0.55 the game chance is 69% and 75%. (b) The same for the match: 84% for best of three to 21, 82% for best of five to 11.

Example research question

To adapt, never to copy

For a table-tennis game to 11 (win by two), how does the chance of winning a game and a best-of-five match depend on the chance of winning a rally, and how well does the model, checked by simulation, predict the results of real matches in one league?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the sport and its scoring rules: points to win, win-by-two, a cap, and the number of games in a match.
Says what is assumed about rallies (independent, the same chance) and how that will be tested.
Names what is calculated: game chance, match chance, amplification, or the chance needed for a given result.
Says how the formula is checked: simulation, real scores, or both.
Goes beyond a table of results by deriving a formula or comparing two rule sets.

Which course it fits

AA SL good
AA HL ideal
AI SL good
AI HL ideal

Syllabus links in Maths AA:

AA 1.3 Geometric sequences and series
AA 1.8 Infinite geometric series
AA 1.9 Binomial theorem
AA 4.5 Basic probability and expected frequency
AA 4.8 Binomial distribution
AA 5.6 Standard derivatives; chain, product and quotient rules

Steps

Read and predict. Find the scoring rules of your sport. Before calculating, guess the chance of winning a game if you win 55% of the rallies.
Fix the rules. Write down points to win, win-by-two, any cap, and games per match. Say which sport uses which rules.
Count the ways. Work out the chance of winning to 11 with the opponent on 0, 1, ... 9, and add them up. Check small cases by listing.
Trial run (1 lesson). Simulate 1,000 games in a spreadsheet or Python. Does the result match your first formula? Fix any mismatch now.
Solve the deuce stage. Find the chance of winning from 10-10 and add it into the game formula. Check it against the simulation.
Build the match. Use the game chance to find the match chance for your format, and graph all three levels against p.
Test on real scores. Estimate a rally chance from real scores and compare the predicted and the actual game and match results.
Go further, with a reason. Differentiate at p = 0.5, or compare two rule sets. HL: this is where the rigour for criterion E comes from.
Reflect as you go. After each result, write one line on what it means for a real player and what it assumes. These lines become your evaluation.
Cut, check, then send. Check every symbol is defined, the simulation agrees with the formula and the conclusion answers your aim. Then send your teacher one draft.

Make it yours

Game to 11 against game to 21. Compute the chance of winning a game under both scoring lengths for the same rally chance.
The deuce stage. Find the chance of winning from 10-10 when you must win by two points.
Amplification factor. Differentiate the game and match win chances with respect to p at p = 0.5.
Simulate it. Simulate thousands of games and matches and compare the results with the formulas.
Test on real scores. Use scores from a league or tournament to estimate a rally chance for each player, then predict match results.
Change the rules. Compare traditional and rally scoring, a point cap, or first to 15 against first to 11.

HL extension

Derive the game formula including the win-by-two stage in closed form, differentiate $M(G(p))$ at $p = 0.5$ by the chain rule, and model the serve advantage with two rally chances, comparing with real scores.

The data you need

Simulation for checking the formulas, plus real scores from your own matches or a published league or tournament to estimate a rally chance and test the predictions.

Traps

Assuming the chance of winning a rally is the same for the server and the receiver. In many sports it is not.
Forgetting the win-by-two rule, or applying it from the start instead of from the tied score.
Using the 21-point formula for badminton without saying that the real game stops at 30.
Applying the model to pickleball's traditional side-out scoring, where only the server can score.
Simulating a few hundred games and calling the formula confirmed.
A simulation that has the same mistake as the formula: check one case by hand.

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