IA idea · Matrices, transformations & Markov chains

Gambler's ruin: how long until someone runs out?

AI HLAA HL Ambitious Also in: Probability, Simulation

Research question

If two players repeatedly bet one coin on a slightly unfair game, what is the probability that each is ruined and how long does the game last on average, from matrices, from a recurrence and from simulation?

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

Three methods for the same answer that check each other: an absorbing Markov chain, a difference equation solved exactly, and a simulation.

The mathematics you'll need

Absorbing Markov chains and the fundamental matrix (new)
Recurrence relations solved with a characteristic equation
Expected duration
Simulation to check
Effect of a small bias over many bets

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 play a real coin game to compare.

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

Set up the chain for small totals.
Find ruin probabilities with matrices.
Solve the recurrence for general totals.
Find expected duration and check by simulation.
Reflect on what a small bias does in a casino.

Pitfalls that cost marks

Using the fundamental matrix without explaining it.
Algebra errors in the recurrence; check with small cases.
Too few simulations.

Showing personal engagement

Predict the answer for a fair game first.
Find the bias that makes a game 'feel' fair but isn't.
Play the game with a friend.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Absorbing Markov chains and the fundamental matrix (new); Recurrence relations solved with a characteristic equation
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	Good fit	Absorbing Markov chains and the fundamental matrix (new); Recurrence relations solved with a characteristic equation

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Collect the data for your matrix yourself (counting transitions, measuring a shape), and choose the states or the transformation from a situation you care about.

Reflection (D)

Question the model's assumptions: is the process memoryless, are the probabilities constant, does the transformation preserve what it should? Say how each affects your conclusion. For this idea, start with: using the fundamental matrix without explaining it — say how it affects your answer.

Use of mathematics (E)

SL: Matrices are in AI HL. At SL, keep to small matrices you explain carefully as new mathematics, with every multiplication shown once and the result interpreted.

HL: Transition matrices, powers, steady states and eigenvalues used correctly, with diagonalisation or a general result derived, and the long-run behaviour interpreted in context.

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

Allow bets of different sizes, or find the strategy that maximises the chance of reaching a target.

Extending it for HL

Diagonalise the matrix to find a formula for the nth state, or compare the steady state with what your data actually shows.

See a complete IA, marked

Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) asks a different question, but shows how a complete matrices & markov 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).