Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

The Monty Hall problem when the host is forgetful

AA SLAA HLAI SLAI HL Solid Also in: Games & puzzles

Research question

How does the probability of winning by switching change if the host opens a door at random (and might reveal the car), or if there are n doors and the host opens k of them?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

The standard Monty Hall answer (switch: 2/3) depends entirely on the host's behaviour. Changing that behaviour shows why conditional probability is subtle — and makes the classic problem your own.

The mathematics you'll need

Conditional probability and Bayes' theorem (AA HL)
Tree diagrams
General formulas for n doors and k opened
Simulation to verify

Where the data comes from

Play the game with cards many times with a friend acting as host; simulate the variants.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Solve the classic version carefully.
Solve the random-host version and explain the difference.
Generalise to n doors and k opened.
Verify with physical trials and simulation.
Reflect on why people's intuition fails.

Pitfalls that cost marks

Only doing the classic version.
Conditioning incorrectly in the random-host case.
Too few physical trials to show anything.

Showing personal engagement

Survey friends' intuition before and after showing your results.
Design your own variant with a biased host.
Explain it in a way a younger student understands.

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	Tree diagrams; General formulas for n doors and k opened
AA HL	Good fit	Conditional probability and Bayes' theorem; Tree diagrams
AI SL	Good fit	Tree diagrams; General formulas for n doors and k opened
AI HL	Good fit	Tree diagrams; General formulas for n doors and k opened

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

Reflection (D)

For this idea, start with the weak points: only doing the classic version; and conditioning incorrectly in the random-host case. Say how each one affects your answer.

Use of mathematics (E)

SL: tree diagrams; general formulas for n doors and k opened; simulation to verify, each explained in your own words and carried out correctly.

HL adds: conditional probability and Bayes' theorem, each explained in your own words and carried out correctly.

How probability explorations reach the top bands in general: Probability & chance IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Treat the host's strategy as unknown and use Bayes' theorem to update your belief after several games.

Extending it for HL

This idea already has HL mathematics in it: Conditional probability and Bayes' theorem. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) is a complete probability exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Conditional probability and Bayes' theorem (AA HL), tree diagrams, general formulas for n doors and k opened and simulation to verify.

Where can I get data for this IA?

Play the game with cards many times with a friend acting as host; simulate the variants. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “The Monty Hall problem when the host is forgetful” filled in: the question, the maths and the data sources.

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