Probability · Maths EE idea · Solid
How long to collect the whole set?
A research question to start from
What is the expected number of purchases needed to collect all n different cards, and how does the answer change if some cards are rarer?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A clean expectation result with a harmonic-series twist and a harder unequal case.
Mathematics you would need
- Geometric distributions
- Linearity of expectation
- Harmonic numbers and ln n
- Variance
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Derive the expectation using geometric waiting times.
- Approximate with ln n and test the approximation.
- Treat unequal probabilities with inclusion–exclusion or integrals.
Scope and difficulty
Solid. Solid.
Pitfalls
- Context about sticker albums taking over.
- Assuming independence wrongly.
Where to start reading
Search a library catalogue or a university's open lecture notes for: coupon collector expected value harmonic; unequal probabilities coupon collector. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).