How many packs does it take to finish a sticker album?

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

Research question

For an album of n stickers sold in packs of k, what is the expected number of packs needed to complete it, and how much does swapping with friends reduce the cost?

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

Why it makes a good exploration

Every football tournament brings a sticker album, and completing it is famously expensive. The coupon collector's problem gives an exact answer involving the harmonic series — then swapping turns it into a question about cooperation.

The mathematics you'll need

Expected value of geometric distributions
Harmonic numbers and the approximation n ln n
Simulation for packs without duplicates and for swapping groups
Cost modelling

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

Use the published sticker count and pack size of a real album, and record your own (or a friend's) packs as test data.

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

Explain the problem and assumptions (all stickers equally likely).
Derive the expected number of stickers needed.
Convert to packs and cost.
Simulate swapping groups of 2, 5 and 10 collectors.
Reflect on whether stickers are really equally likely.

Pitfalls that cost marks

Assuming equally likely stickers without testing.
Ignoring that packs have no duplicates within them.
Stating n ln n without derivation.

Showing personal engagement

Collect data from your own album.
Test whether rare stickers exist with a chi-squared test.
Calculate when buying missing stickers directly becomes worthwhile.

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

Taking it further

Derive the variance of the number of stickers needed and discuss how unlucky a collector can be.

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).