IA idea · Probability & chance
How many packs does it take to finish a sticker album?
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?
Make it yours, then agree it with your teacher.
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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
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.
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.
Which course is it for?
| Course | Fit | Maths to lean on |
|---|---|---|
| AA SL | Good fit | Expected value of geometric distributions; Harmonic numbers and the approximation n ln n |
| AA HL | Good fit | Expected value of geometric distributions; Harmonic numbers and the approximation n ln n |
| AI SL | Good fit | Expected value of geometric distributions; Harmonic numbers and the approximation n ln n |
| AI HL | Good fit | Expected value of geometric distributions; Harmonic numbers and the approximation n ln n |
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: assuming equally likely stickers without testing; and ignoring that packs have no duplicates within them. Say how each one affects your answer.
Use of mathematics (E)
SL: expected value of geometric distributions; harmonic numbers and the approximation n ln n; simulation for packs without duplicates and for swapping groups; cost modelling, 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
Derive the variance of the number of stickers needed and discuss how unlucky a collector can be.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL probability exploration adds.
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?
Expected value of geometric distributions, harmonic numbers and the approximation n ln n, simulation for packs without duplicates and for swapping groups and cost modelling.
Where can I get data for this IA?
Use the published sticker count and pack size of a real album, and record your own (or a friend's) packs as test data. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “How many packs does it take to finish a sticker album?” filled in: the question, the maths and the data sources.
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