When should you stop looking? The 37% rule for choosing the best

AA HLAA SLAI HL Ambitious Also in: Pure maths, Games & puzzles

Research question

In the secretary problem, why is rejecting the first n/e candidates the optimal strategy, and how well does the rule work in a realistic version such as house hunting with a limited number of viewings?

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

Why it makes a good exploration

Whether choosing a flat, a university or a parking space, you must decide without knowing what comes next. The optimal stopping rule is surprising and elegant, and testing it on a realistic scenario gives excellent Reflection.

The mathematics you'll need

Conditional probability and summation
Approximating a sum by an integral to get 1/e
Maximising a function of the cut-off
Simulation for finite n and variants

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 for the core; simulate variants in a spreadsheet or short program.

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

State the problem and assumptions.
Derive P(success) for a cut-off r.
Find the optimal r for small n exactly, then the limit n/e.
Simulate variants (recall allowed, aiming for top 3).
Reflect on real decisions that break the assumptions.

Pitfalls that cost marks

Quoting 37% without derivation.
Not checking small cases by hand.
Ignoring that the rule maximises the chance of the best, not the average quality.

Showing personal engagement

Apply it to a real decision you face.
Run the experiment on classmates with shuffled cards.
Design a version that optimises expected rank instead.

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

Taking it further

HL: derive the result with the integral approximation carefully and investigate the “full information” version where values are known.

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).