## Got, got, need: how much does a 2014 World Cup sticker collection cost, or why is swapping so important?

The year 2014 brings with it a World Cup, and with that an activity that is familiar to many: the collection of stickers in the attempt to complete Panini’s World Cup sticker album. The pastime has not changed much over the years since Panini’s first collection for the 1970 World Cup. The Italian company still prints the same multilingual booklet, with the pages for most, but not all, of the teams headed with the name of that country in its official language – for Belgium they split the difference and go with ‘Belgique/België’ (sorry German-speaking Belgians!), but rather than entertain a four-way tie they plump with ‘Switzerland’. The major change though is that the number of stickers to collect has increased, largely due to the World Cup itself having expanded to 32 teams in 1998. This year’s collection comprises 640 stickers: 19 for each team (17 players, the team and the logo) with the remaining stickers being the stadia, trophy, official ball etc. So how much does this cost?

The good news is that the empty album can be picked up for free, and furthermore comes with an insert of 6 stickers: Yaya Touré (Côte d’Ivoire), Cristiano Ronaldo (Portugal), Hugo Lloris (France), Gonzalo Higuaín (Argentina), Daniele  De Rossi (Italy) and Roman Shirokov (Russia). This leaves 634 stickers to buy. Stickers can be bought in packs of 5 for 50p, but you get multipacks of 5 packs for £2 in Asda. So 25 multipacks and 2 extra packs would get you 640 stickers at a cost of £51 (just buying individual packs this cost would be £63.50.)

But, of course, among these stickers are likely to be duplicates. If I were to hand you 636 randomly selected stickers the chances you’d have a complete collection would be about 1 in $10^{276}$. So how much would we expect to spend before we have a complete collection. We’ll assume that each sticker is equally likely to appear (even the ones that come in the insert), so each sticker has a $\frac{1}{640}$ probability of being Yaya Touré, or of being Wayne Rooney, or of being the Colombian logo.

So far we already have 6 of our album stickers (as stickers stuck in our album will be referred to hereon in), so when we open the first pack the first sticker we see could be one of 634 we don’t already have or one of the 6 we do, so the probability it is our 7th album sticker is $\frac{634}{640}$, while the probability it is a duplicate is $\frac{6}{640}$. If it is a duplicate then the next sticker still has a probability of $\frac{634}{640}$ of being our 7th album sticker, and this remains the case until we find our 7th album sticker. The number of stickers we see until we have our 7th album sticker is equivalent to the number of success/failure (i.e. Bernoulli) trials until the first success. The number of Bernoulli trials of success probability $p$ until success follows a Geometric distribution and its mean value is simply given by $\frac{1}{p}$. So we expect to see $\frac{640}{634}$ stickers until we have our 7th album sticker.

Once we have our 7th album sticker, we have now 633 stickers left to collect. So the probability of a sticker being our 8th album sticker is $\frac{633}{640}$. After our 7th album sticker, the number of stickers we see until we have our 8th album sticker again follows a Geometric distribution, so we expect to see $\frac{640}{633}$ more stickers until we have our 8th album sticker. We then expect to see $\frac{640}{632}$ stickers until we see our 9th album sticker, and $\frac{640}{631}$ until we see our 10th and so on.

So, once we have $latex i-1$ stickers, the number of stickers we expect to see before we find our $i$th album sticker is $\frac{640}{641-i}$. All told then the number of stickers we’d expect to see to complete our album of 640 stickers is

$\sum_{i=7}^{640}\frac{640}{641-i}=\frac{640}{634}+\frac{640}{633}+\frac{640}{632}+\cdots+\frac{640}{2}+\frac{640}{1}=4500,$

which would be 180 multipacks at a total cost of £360! (Or 900 individual packs at a cost of £450.)

The main reason for this high cost is that as you fill up your sticker album, it becomes harder and harder to add to it. Once you have 639 stickers, you’d expect to have to see 640 stickers before you get your final sticker, at a cost of £51.20 (at the 8p per sticker multipack price) or £64 (at the 10p per sticker individual pack price), which is a lot of money to get one sticker! You can see the expected number of stickers and cost at each rate for each album sticker in the spreadsheet below (the would-be cost of album stickers is included – their being free doesn’t offer a huge saving.)

Thankfully Panini have mercy and allow you a one-off order of 50 specified stickers to help you complete your collection, at the price of 14p per sticker plus £1 for postage. This works out at 16p per sticker if you order 50, which reduces to 14.6p with a 10% discount on the stickers if you order online using a code in your album. While this is more expensive than the cost of a sticker it’s still much cheaper than the cost of later album stickers, in fact from the spreadsheet we can see that even with multipacks each new album sticker is expected to cost over 14.6p when you only have 290 album stickers – not even half full! Ordering these stickers is a no-brainer and requires us then to only collect 590 album stickers beforehand, the number of stickers we’d expect to see to reach this amount is

$\sum_{i=7}^{590}\frac{640}{641-i}=1619,$

which requires 65 multipacks at a total cost of £130 or 324 individual packs at a total cost of £162. Add in your ordered album stickers at the reduced rate and you’re expecting to pay £137.30 with multipacks or £169.30 with individual packs, a significant reduction thanks to being able to order those last 50 otherwise incredibly expensive stickers. It is still however a good deal more than the £51 minimum cost – which is what makes the ability to make good swaps so vital, to reduce the expected cost closer towards that lower value.