IA idea · Simulation & Monte Carlo methods

Does the best team win? Knockout cups versus leagues

AI SLAI HLAA SLAA HL Solid Also in: Sport, Voting & game theory, Probability

Research question

Using match-win probabilities estimated from real results, how often does the strongest team win a knockout cup compared with a round-robin league, and how many rounds or legs does a cup need to be as reliable as a league?

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

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Fans argue about whether cups are 'lotteries'. Estimating win probabilities from real data and simulating thousands of tournaments gives a numerical answer and a fair comparison of formats.

The mathematics you'll need

Estimating match probabilities from results
Exact probability for small knockout brackets (tree diagrams)
Simulation of full tournaments
Comparing proportions; HL: confidence intervals
Effect of seeding and two-legged ties

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.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Confidence intervals for a mean. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Download a season of results from Football-Data.co.uk or a sport you follow and estimate team strengths.

Football-Data.co.uk — Match results, shots, cards and bookmaker odds for 25+ seasons of European leagues, CSV.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

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

Estimate win, draw and loss probabilities for each pairing from data.
Calculate exactly for a four-team knockout.
Simulate an eight- or sixteen-team cup and a league.
Compare how often the best team wins each format.
Reflect on the probability model and home advantage.

Pitfalls that cost marks

Probabilities that ignore draws or home advantage without comment.
Too few simulated tournaments.
No exact check for a small case.

Showing personal engagement

Use your own league or sport.
Ask whether your school tournament format is fair.
Design the fairest format that fits in one afternoon.

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	Estimating match probabilities from results; Exact probability for small knockout brackets (tree diagrams)
AA HL	Good fit	Estimating match probabilities from results; Exact probability for small knockout brackets (tree diagrams)
AI SL	Good fit	Estimating match probabilities from results; Exact probability for small knockout brackets (tree diagrams)
AI HL	Good fit	Estimating match probabilities from results; Exact probability for small knockout brackets (tree diagrams)

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

Personal engagement (C)

Design the simulation yourself: choose the rules, test them on a small case you can check by hand, and change one rule at a time to answer a question you care about.

Reflection (D)

Compare simulation with exact theory or real data, say how many runs you used and how much the answer varies between batches, and question the random-number assumptions. For this idea, start with: probabilities that ignore draws or home advantage without comment — say how it affects your answer.

Use of mathematics (E)

SL: A probability model described precisely, simulated correctly, with the simulated answer compared with an exact calculation for at least one simple case and the results summarised with appropriate statistics.

HL: An estimate of the simulation's error (standard error, or a confidence interval for the estimate), a distribution fitted to the results and tested, or an exact result proved for the general case that the simulation confirms.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Model team strengths with an Elo rating and simulate a whole season, or find the format that maximises excitement as well as fairness.

Extending it for HL

Give a confidence interval for each simulated estimate and show how it narrows as the number of runs grows, or prove a general result that the simulation confirms.

See a complete IA, marked

Our annotated exemplar How long does a skydiver take to reach terminal velocity? (AA HL) asks a different question, but shows how a complete simulation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).