IA idea · Voting, fairness & game theory

Secret Santa: how often does someone draw their own name?

AA HLAA SL Solid Also in: Probability, Simulation

Research question

What is the probability that nobody draws their own name in a Secret Santa, why does it approach 1/e, and is the common 'redraw if you get yourself' rule fair to everyone?

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

Derangements are a classic counting result, and the fairness question about real redraw rules is a genuine new angle you can test by simulation.

The mathematics you'll need

Counting permutations and derangements
Recurrence for derangements
Inclusion–exclusion (new)
The limit 1/e via the series for e
Simulation of redraw rules

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 record a real draw.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

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

Count derangements for small n by listing.
Find and prove a recurrence.
Derive the probability and its limit.
Simulate common redraw rules.
Reflect on which rule is fair.

Pitfalls that cost marks

Quoting the formula without deriving it.
Simulating a different procedure from the real one.
No definition of fairness for the redraw rule.

Showing personal engagement

Use your family's or club's real rules.
Predict the probability first.
Design a fair procedure.

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	Counting permutations and derangements; Recurrence for derangements
AA HL	Good fit	Counting permutations and derangements; Recurrence for derangements
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)

Run a real vote or game with people you know, and choose the methods or rules to compare. Predict the outcome before you analyse it.

Reflection (D)

Reflect on the gap between the mathematically rational choice and what people did, and on what each fairness method gains and gives up. For this idea, start with: quoting the formula without deriving it — say how it affects your answer.

Use of mathematics (E)

SL: Each method explained with a worked example, then analysed with probability, expected value or counting; results compared systematically rather than case by case.

HL: Mixed strategies found by solving equations or with calculus, a proof that a method has (or lacks) a fairness property, or a probability model of how often methods disagree.

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 probability that the draw forms a single loop.

Extending it for HL

Prove a fairness property in general, or model random ballots and calculate how often two methods pick different winners.

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 voting & game theory 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).