H044
How many stickers does it take to fill a 670-sticker World Cup album?
The album fills up fast at first. Then almost every new sticker is one you already have.
Fast start, slow finish
Average progress when every sticker is equally likely and nothing is traded
The more of a set you already have, the longer each new piece takes to find. With the first sticker every one is new. With 669, only 1 in 670 is. Adding up the waits gives about 4,750 stickers on average, roughly 950 packs of five. Half the album comes in the first 464 stickers; the last 10 take about 1,960 more. Mathematicians call this the coupon collector's problem.
It is why trading works so well: swapping duplicates turns the slow, lonely end of the curve into a shared one. The same pattern shows up whenever you need "one of each": collecting every card in a set, hearing from every person on a list, or testing every case in a system.
Try it somewhere else
A teacher wants every one of 30 students to answer at least once, but calls on a random student each time.
About how many questions will it take before everyone has had a turn?
The math, and the answer for the class
To collect all n items, the expected number of tries is n × (1 + 1/2 + 1/3 + ... + 1/n). For 670 stickers that is about 670 × 7.09 ≈ 4,750. For the class of 30, it is about 30 × 4.0 ≈ 120 questions, four times the number of students. Real albums can be a bit worse if some stickers are printed less often, and a bit better since a single pack has no duplicates.
Related exhibits
Sources: Album size: Panini FIFA World Cup Qatar 2022 sticker album (670 stickers). Expected counts computed with the coupon collector's formula, assuming every sticker is equally likely and no trades; see Sardy & Velenik, "Paninimania: sticker rarity and cost-effective strategy," University of Geneva, 2010.