IA idea · Simulation & Monte Carlo methods

What is a 'random' chord? Bertrand's paradox simulated

AA HLAA SL Solid Also in: Geometry & Voronoi, Probability

Research question

What is the probability that a random chord of a circle is longer than the side of the inscribed equilateral triangle, and why do three reasonable ways of choosing a chord give three different answers?

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

The paradox shows that 'random' needs defining. You can calculate all three answers exactly, simulate them, and then argue (or test physically) which definition fits a real process.

The mathematics you'll need

Chord length from geometry and trigonometry
Three probability models: random endpoints, random radius point, random midpoint
Exact probabilities (1/3, 1/2, 1/4) derived
Simulation of each model
Distributions of chord length

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

No data needed; optionally drop straws on a circle drawn on paper to see which model a physical process follows.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
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

Derive the threshold length.
Calculate the probability under each model.
Simulate each and compare.
Run a physical experiment and decide which model it matches.
Reflect on what 'random' means in probability.

Pitfalls that cost marks

Presenting the three answers without deriving them.
Simulating a different model from the one you think you are simulating.
Ignoring the physical experiment's own biases.

Showing personal engagement

Do the straw-drop experiment yourself.
Invent a fourth way to choose a chord and calculate its answer.
Explain the paradox to a classmate and record their intuition.

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	Chord length from geometry and trigonometry; Three probability models: random endpoints, random radius point, random midpoint
AA HL	Good fit	Chord length from geometry and trigonometry; Three probability models: random endpoints, random radius point, random midpoint
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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: presenting the three answers without deriving them — 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

Find the full probability distribution of chord length under each model, or explore the 'maximum ignorance' argument for one of them.

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 What is the quickest way to deliver a leaflet to every house on my estate? (AI 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

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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).