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2018-06-16

## World Cup stickers 2018, pt. 2

This year, like every World Cup year, I've been collecting stickers to fill the official Panini World Cup sticker album. Back in March, I calculated that I should expect it to cost £268.99 to fill this year's album (if I order the last 50 stickers). As of 6pm yesterday, I need 47 stickers to complete the album (and have placed an order on the Panini website for these).

### So... How much did it cost?

In total, I have bought 1781 stickers (including the 47 I ordered) at a cost of £275.93. The plot below shows the money spent against the number of stickers stuck in, compared with the what I predicted in March.
To create this plot, I've been keeping track of exactly which stickers were in each pack I bought. Using this data, we can look for a few more things. If you want to play with the data yourself, there's a link at the bottom to download it.

### Swaps

The bar chart below shows the number of copies of each sticker I got (excluding the 47 that I ordered). Unsurprisingly, it looks a lot like random noise.
The sticker I got most copies of was sticker 545, showing Panana player Armando Cooper.
Armando Cooper on sticker 545
I got swaps of 513 different stickers, meaning I'm only 169 stickers short of filling a second album.

### First pack of all swaps

Everyone who has every done a sticker book will remember the awful feeling you get when you first get a pack of all swaps. For me, the first time this happened was the 50th pack. The plot below shows when the first pack of all swaps occurred in 500,000 simulations.
Looks like I was really quite unlucky to get a pack of all swaps so soon.

### Duplicates in a pack

In all the 345 packs that I bought, there wasn't a single pack that contained two copies of the same sticker. In fact, I don't remember ever getting two of the same sticker in a pack. For a while I've been wondering if this is because Panini ensure that packs don't contain duplicates, or if it's simply very unlikely that they do.
If it was down to unlikeliness, the probability of having no duplicates in one pack would be:
\begin{align} \mathbb{P}(\text{no duplicates in a pack}) &= 1 \times\frac{681}{682}\times\frac{680}{682}\times\frac{679}{682}\times\frac{678}{682}\\ &= 0.985 \end{align}
and the probability of none of my 345 containing a duplicate would be:
\begin{align} \mathbb{P}(\text{no duplicates in 345 packs}) &= 0.985^{345}\\ &= 0.00628 \end{align}
This is very very small, so it's safe to conclude that Panini do indeed ensure that packs do not contain duplicates.

### The data

If you'd like to have a play with the data yourself, you can download it here. Let me know if you do anything with it...

### Similar posts

 World Cup stickers 2018 World Cup stickers Euro 2016 stickers A bad Puzzle for Today

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2018-03-22

## World Cup stickers 2018

Back in 2014, I worked out the cost of filling an official Panini World Cup 2014 sticker book. Today, the 2018 sticker book was relased: compared to four years ago, there are more stickers to collect and the prices have all changed. So how much should we expect it to cost us to fill the album this time round?

### How many stickers will I need?

There are 682 stickers to collect. Imagine you have already stuck $$n$$ stickers into your album. The probability that the next sticker you buy is new is
$$\frac{682-n}{682}.$$
The probability that the second sticker you buy is the next new sticker is
$$\mathbb{P}(\text{next sticker is not new})\times\mathbb{P}(\text{sticker after next is new})$$ $$=\frac{n}{682}\times\frac{682-n}{682}.$$
Following the same method, we can see that the probability that the $$i$$th sticker you buy is the next new sticker is
$$\left(\frac{n}{682}\right)^{i-1}\times\frac{682-n}{682}.$$
Using this, we can calculate the expected number of stickers you will need to buy until you find a new one:
$$\sum_{i=1}^{\infty}i \left(\frac{682-n}{682}\right) \left(\frac{n}{682}\right)^{i-1} = \frac{682}{682-n}$$
Therefore, to get all 682 stickers, you should expect to buy
$$\sum_{n=0}^{681}\frac{682}{682-n} = 4844 \text{ stickers}.$$

### How much will this cost?

You can buy the following [source]:
• Starter pack (an album and 31 stickers) for £3.99
• Sticker packs (5 stickers) for 80p
• Sticker multipacks (30 stickers) for £4.50
First of all you'll need to buy the starter pack, as you need an album to stick everything in. This comes with 31 stickers; we should expect to buy 4813 more stickers after this.
The cheapest way to buy these stickers is to buy them in multipacks for 15p per sticker. This gives a total expected cost of filling the sticker album of £725.94. (Although if your local newsagent doesn't stock the multipacks, buying 80p packs to get these stickers will cost you £774.07.)

### What if I order the last 50 stickers?

If you'd like to spend a bit less on the sticker book, Panini lets you order the last 50 stickers to complete your album. This is very helpful as these last 50 stickers are the most expensive.
You can order missing stickers from the Panini website for 22p per sticker's sticker ordering service for the 2018 World Cup doesn't appear to be online yet; but based on other recent collections, it looks like ordered stickers will cost 16p each, with £1 postage per order.
Ordering the last 50 stickers reduces the expected number of other stickers you need to buy to
$$\sum_{n=0}^{631}\frac{682}{682-n} = 1775 \text{ stickers}.$$
This reduces the expected overall cost to £291.03. So I've just saved you £434.91.

### What if I order more stickers?

Of course, if you're willing to completely give up on your morals, you could order more than one batch of 50 stickers from Panini. This raises the question: how many should you order to minimise the expected cost.
If you order the last $$a$$ stickers, then you should expect to pay:
• £3.99 for the album and first 31 stickers
• £$$\displaystyle0.15\left(\sum_{n=1}^{681-a}\frac{682}{682-n}-31\right)$$ for other stickers you need
• £$$\displaystyle \left(0.16a+\left\lceil \frac{a}{50}\right\rceil\right)$$ to order the last $$a$$ stickers
The total expected cost of filling your album for different values of $$a$$ is shown in the graph below.
The red cross shows the point at which the album is cheapest: this is when the last 550 stickers are ordered, giving a total expected cost of £120.18. That's another £170.85 I've saved you. You're welcome.
Still, ordering nearly all stickers to minimise the cost doesn't sound like the most fun way to complete the sticker book, so you probably need to order a few less than this to maximise your fun.

Of course, you can also get the cost of filling the book down by swapping your spare stickers with friends. In 2016, I had a go at simulating filling a sticker book with swapping and came to the possibly obvious conclusion that the more friends you have to swap with, the cheaper filling the book will become.
My best advice for you, therefore, is to get out there right now and start convinding your friends to join you in collecting stickers.

### Similar posts

 World Cup stickers 2018, pt. 2 World Cup stickers Euro 2016 stickers A bad Puzzle for Today

Comments in green were written by me. Comments in blue were not written by me.
2018-03-24
It would slightly reduce the expected cost I think, but I need to do some calculations to make sure.

I'm not certain whether they guarantee that there are no duplicates in a pack or just it's very unlikely... Follow up post looking into this coming soon
Matthew
2018-03-23
Helpful article!! My head always go into a spine with probability, but does the fact that you would not get a double within a pack affect the overall strategy or cost? My head says this is an another advantage of buying 30 packs over the 5.
RM

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2014-05-26

## World Cup stickers

With the FIFA World Cup approaching, sticker fans across the world are filling up their official Panini sticker books. This got me wondering: how many stickers should I expect to need to buy to complete my album? And how much will this cost?

### How many stickers?

There are 640 stickers required to fill the album. The last 100 stickers required can be ordered from the Panini website.
After $$n$$ stickers have been stuck into the album, the probability of the next sticker being the next new sticker is:
$$\frac{640-n}{640}$$
The probability that the sticker after next is the next new sticker is:
$$\frac{n}{640}\frac{640-n}{640}$$
The probability that the sticker after that is the next new sticker is:
$$\left(\frac{n}{640}\right)^2\frac{640-n}{640}$$
Following this pattern, we find that the expected number of stickers bought to find a new sticker is:
$$\sum_{i=1}^{\infty}i \left(\frac{640-n}{640}\right) \left(\frac{n}{640}\right)^i = \frac{640}{640-n}$$
Therefore, to get all 640 stickers, I should expect to buy:
$$\sum_{n=0}^{639}\frac{640}{640-n} = 4505 \mbox{ stickers.}$$
Or, if the last 100 stickers needed are ordered:
$$\sum_{n=0}^{539}\frac{640}{640-n} + 100 = 1285 \mbox{ stickers.}$$

### How much?

The first 21 stickers come with the album for £1.99. Additional stickers can be bought in packs of 5 for 50p or multipacks of 30 for £2.75. To complete the album, 100 stickers can be bought for 25p each.
If I decided to complete my album without ordering the final stickers, I should expect to buy 4505 stickers. After the 21 which come with the album, I will need to buy 4484 stickers: just under 897 packs. These packs would cost £411.25 (149 multipacks and 3 single packs), giving a total cost of £413.24 for the completed album.
I'm not sure if I have a spare £413.24 lying around, so hopefully I can reduce the cost of the album by buying the last 100 stickers for £25. This would mean that once I've received the first 21 stickers with the album, I will need to buy 1164 stickers, or 233 packs. These packs would cost £107 (38 multipacks and 5 single packs), giving a total cost of £133.99 for the completed album, significantly less than if I decided not to buy the last stickers.

### How many should I order?

The further reduce the number of stickers bought, I could get a friend to also order 100 stickers for me and so buy the last 200 stickers for 25p each. With enough friends the whole album could be filled this way, although as the stickers are more expensive than when bought in packs, this would not be the cheapest way.
If the last 219 or 250 stickers are bought for 25p each, then I should expect to spend £117.74 in total on the album. If I buy any other number of stickers at the end, the expected spend will be higher.
Fortunately, as you will be able to swap your duplicate stickers with your friends, the cost of a full album should turn out to be significantly lower than this. Although if saving money is your aim, then perhaps the Panini World Cup 2014 Sticker Book game would be a better alternative to a real sticker book.

### Similar posts

 World Cup stickers 2018, pt. 2 World Cup stickers 2018 Euro 2016 stickers How to kick a conversion