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?

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

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

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

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.

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

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.

Taking it further

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

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).