I don’t watch the World Cup, or sports in general. I never get excited about it, and I certainly do not feel any affinity towards any teams.

I think instead of living vicariously through others doing what they like, we should just do what we really want.

Anyway, my good friend Sawan’s little sister recently a box of 1000 World Cup stickers, which cost a whopping £150. According to him, the whole point is that you are meant to collect your dupes as they call it and trade them with other peoples dupes to fill in your last missing stickers. If you try to finish the book without trading you’ll end up spending thousands.

So he asked me the following question:

The FIFA World Cup sticker album contains 670 unique stickers, each corresponding to a different player. Stickers are distributed uniformly at random, so every sticker purchased is equally likely to be any one of the 670 designs, with replacement.

Suppose you purchase a Mega Box containing 1,000 stickers (at a cost of £150).

Question: On average, how many of the 1,000 stickers would you expect to be duplicates?

More generally, if an album contains $N$ unique stickers and you purchase $n$ stickers at random, can we derive a formula for the expected number of duplicate stickers?

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Once again, this is a good moment to pause the article and give the problem a go yourself. When you’re ready, keep reading for the solution. And if you come up with your own approach, feel free to share it in the comments — I’d love to see how you tackled it.

Don’t forget to subscribe to our YouTube channel for more maths puzzles, it’s my goal to reach 100k subscribers someday. 

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Solution

The key idea is to calculate the expected number of different stickers you get after opening 1,000 stickers.

Step 1: Focus on one particular sticker

Pick one specific sticker, say sticker #42.

The probability that a single draw is not sticker #42 is

because there are 669 other stickers it could be.

Step 2: What is the probability you never get it?

Since you open 1,000 stickers independently, the probability that sticker #42 never appears is

Therefore, the probability that you get sticker #42 at least once is

Step 3: Extend to all 670 stickers

Each of the 670 sticker types has the same probability of appearing at least once.

The expected number of unique stickers is simply

This works because of linearity of expectation: if each sticker has probability $p$ of appearing at least once, then the expected number that appear is just $670\mathrm{p.}$

$670p$Step 4: Calculate it

First,

So

Then we multiply by 670

So we get around 520 unique stickers.

Step 5: Find Duplicates

You opened 1,000 stickers total, so we subtract the number of unique stickers to get expected number of duplicates.

Bonus: More generally, if an album contains $N$ unique stickers and you purchase $n$ stickers at random, can we derive a formula for the expected number of duplicate stickers?

And that’s our answer. How amazing. Hey, don’t forget to clap👏 the article as a token of appreciation. Thank you 🦁

$$1000-520\approx480.$$