Maths IA · Statistics and probability · topic 24 of 40

Tie-breaks: do league rules change who wins? · IB Maths IA idea: guide, sample graph and step-by-step plan

sport + maths: easy to care about

Two teams finish on the same points. Who lifts the trophy? You're going to find out whether the tie-break rule quietly decides the league.

Difficulty●●○ Medium
KitSpreadsheet or Python, GDC
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like sport and league tables
  • ✓ you can code or build spreadsheet formulae
  • ✓ you enjoy testing a claim with thousands of trials

The big idea

← the surprise is your hook

A tie-break is a rule for ending a tie, and different rules (goal difference, head-to-head, goals scored) can crown different champions from the same results. Build a probability model of a small league, simulate thousands of seasons, and measure how often the winner depends on the rule. Work out a tiny league exactly, so your simulation has a true answer to be checked against. The maths is in the model, the counting and how your estimates settle. Strong explorations test the simulation against exact answers instead of trusting big numbers.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Model goals, not just wins

Give each team an attack and a defence strength and generate scores from a Poisson distribution (AI HL; an extension on AA).

→ Goal difference and goals scored now mean something, so the rules can really disagree. You must justify the model and say what it assumes.
Add a rule that is not obvious

Compare head-to-head mini-tables, or a random draw, with the usual rules.

→ More rules to compare, so you need a clear measure of how different two rules are.
Define 'fair' yourself

Decide what a fair rule should do, for example give the strongest team the best chance.

→ Your outcome becomes the chance the best team wins under each rule. You must argue why that is fair.
Shrink the league to count exactly

Three or four teams, one round of matches, so every outcome can be listed.

→ Exact probabilities from counting or a probability tree: a true value to check the simulation against.
Fit the strengths to real data

Use a real season of a small league, or your school or club league, to set the team strengths.

→ Your own data and your own choices, but you must say how much a short season can really tell you.
Test the simulation itself

Repeat the whole simulation at 100, 1 000 and 10 000 seasons.

→ Shows how the error in an estimated proportion shrinks, and lets you give confidence intervals instead of bare numbers.

Your Maths IA research question: run the check

define 'winner' BEFORE you start

Write your own question first. Then make sure it ticks every box:

  • Names the league: how many teams, single or double round-robin, how many matches each.
  • Names the tie-break rules compared and defines each as an ordered list, including what happens if a rule still leaves a tie.
  • Says what you will measure (how often the champion changes, or how often the strongest team wins) and how you will count it.
  • Says how a match result is generated, and what that assumes about the teams.
  • Is narrow enough: a small league and two to four rules, not 'football in general'.
  • Can be answered with numbers that you can quote with a stated precision.

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The model

One possible method as a flow chart. Draw yours with your own model and rules: every box is a decision you will have to justify.

Team strengths(your model)Simulate one seasonrandom resultsFinal table:which teams are level?Rule A:goal differenceRule B:head-to-headRule C:goals scoredChampion tally,repeated 10 000 timesstart small: 4 teamsties are the point!swap ONE ruleat a time

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Title page, question and rationale~1 page
Your question; the league or sport that made you ask it; why a tie-break matters to the people in it.
Aim and plan~0.5 page
What you will measure, which rules you compare and a short route map of the maths.
The model~2 pages
How a match result is generated, what it assumes and why; every symbol defined, with units where they apply.
An exact case~2 pages
A tiny league worked out by counting or a probability tree, so you have true values to test against.
The simulation~2 pages
How one season is run, as a flow chart or steps in your own notation; the number of seasons and why. Long code goes in an appendix.
Results~2 pages
One clear table and one or two graphs of how often each rule changes the champion, with the precision of each estimate.
Testing the estimates~2 pages
Simulation against exact answers; a confidence interval or a test comparing rules, with its conditions checked. At HL this is where the HL-level maths must show.
Reflection throughout~1.5 pages
Where the model is unrealistic (equal chances? independent matches?), what changing it does, and what you would trust. Put some after each stage, not only at the end.
Conclusion~0.5 page
Answer your question with numbers and say whether the aim was met.
Bibliography and appendixnot counted
Sources, long code and bulk output.

What your Maths IA graph could look like

say what each point is made of

Made-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only

69121518212427300.000.020.040.060.080.100.120.140.16Matches played by each team, nProbability of a points tie for first placesimple model: k / √n, k fitted to short seasonssimulated, 2 000 seasons each (4 unequal teams)the points fall below the curvein long seasons: why?ILLUSTRATIVE · MADE-UP DATA
  • Axes name the quantity, with the number of matches as a clear integer scale.
  • Error bars come from the simulation size: say how you worked them out, since they are not measurement errors.
  • The curve is a simple model with a reason (for equal teams the spread of points grows like √n). The data is judged against it, and where they part is a finding.
  • Say how many seasons sit behind each point, so the reader can judge the precision.
  • Another graph to consider: how the estimated proportion settles as the number of simulated seasons grows, with its confidence interval narrowing.

Your game plan

trial run first!

Tick steps off as you go. We'll remember on this device.

0 of 11 done

  1. Pick a league and a puzzleChoose a sport or competition you follow. Find a case where a tie-break mattered, or imagine one. Write down in your own words what you are curious about.
  2. Define the rules exactlyWrite each tie-break as an ordered list. What happens if teams are still level after the last rule? Vague rules ruin everything later.
  3. Predict firstBefore any calculation, say which rule you think will change the winner most, and why.
  4. Trial run (one evening)Do a three-team league by hand, then a tiny simulation of 10 seasons. Do ties happen often enough to study? Does your flow chart actually work?
  5. Choose and justify the modelDecide how results are generated and what you are assuming. Check that your probabilities add to 1 and the outcomes are sensible.
  6. Get an exact answerIn a tiny league, count every outcome or draw a probability tree. This gives you something true to check against.
  7. Scale up and checkRun thousands of seasons. Compare with the exact case, then increase the number of seasons and record how the estimates settle.
  8. Compare the rulesDecide the measure and the test. What would count as a real difference rather than chance variation in a simulation?
  9. Reflect as you goAfter each stage, note what surprised you and what you changed. Reflection belongs in the body of the work.
  10. Write, then cutAim for 12 to 20 pages. Show one worked example and put repeated runs in a table. Explain code in words and mathematics, and keep it for an appendix.
  11. Check before your teacher's draftCheck notation (no computer syntax), axis titles, defined symbols, and that your conclusion answers your aim.

Maths you'll need

Traps to avoid in this Maths IA

  • Reporting simulation output with no check against an exact answer.
  • All teams equally strong: there is no 'best team', so you cannot judge which rule is fairer.
  • Pasting code or spreadsheet screenshots instead of explaining the model in proper notation.
  • Quoting five decimal places from a thousand seasons. State how precise your estimate really is.
  • At HL, counting wins and drawing a bar chart is not enough for criterion E: you need HL-level maths such as distributions, confidence intervals or a hypothesis test, used with care.
  • Rules that are not defined clearly enough for a reader to repeat your work.
  • Treating matches as independent, or teams as constant over a season, without saying so.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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