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2016-05-04 16:15:21

## Euro 2016 Stickers

Back in 2014, I calculated the expected cost of filling Panini world cup sticker album. I found that you should expect to buy 4505 stickers, or 1285 if you order the last 100 from the Panini website (this includes the last 100). This would cost £413.24 or £133.99 respectively.
Euro 16 is getting close, so it's sticker time again. For the Euro 16 album there are 680 stickers to collect, 40 more than 2014's 640 stickers. Using the same calculation method as before, to fill the Euro 16 album, you should expect to buy 4828 stickers (£442.72), or 1400 (£134.32) if you order the last 100.
This, however, does not tell the whole story. Anyone who has collected stickers as a child or an adult will know that half the fun comes from swapping your doubles with friends. Getting stickers this way is not taken into account in the above numbers.

### Simulating a Sticker Collection

Including swaps makes the situation more complicated: too complicated to easily calculate the expected cost of a full album. Instead, a different method is needed. The cost of filling an album can be estimated by simulating the collection lots of times and taking the average of the cost of filling the album in each simulation. With enough simulations, this estimate will be very close the the expected cost.
To get an accurate estimation, simulations are run, calculating the running average as they go, until the running averages after recent simulations are close together. (In the examples, I look for the four most recent running averages to be within 0.01.) The plot below shows how the running average changes as more simulations are performed.
The simulations estimate the number of stickers needed as 4500. This is very close to the 4505 I calculated last year.
Now that the simulations are set up, they can be used to see what happens if you have friends to swap with.

### What Should I Do?

The plots below shows how the number of stickers you need to buy each changes based on how many friends you have.
Stickers needed if you and your friends order no stickers.
Stickers needed if you and your friends all order the last 100 stickers. The last 100 are not counted.
In both these cases, having friends reduces the number of stickers you need to buy significantly, with your first few friends making the most difference.
Ordering the last 100 stickers looks to be a better idea than ordering no stickers. But how many stickers should you order to minimise the cost? When you order stickers, you are guaranteed to get those that you need, but they cost more: ordered stickers cost 14p each, while stickers in 6 pack multipacks come out at just 9.2p each. The next plot shows how the cost changes based on how many you order.
The expected cost of filling an album based on number of people in group and number of stickers ordered.
Each of the coloured curves represents a group of a different size. For each group, ordering no stickers works out the most expensive—this is expected as so many stickers must be bought to find the last few stickers—and ordering all the stickers also works out as not the best option. The best number to order is somewhere in the middle, where the curve reaches its lowest point. The minimum points on each of these curves are summarised in the next plots:
How the number you should order changes with the number of people in the group.
How the cost changes with the number of people in the group.
Again, having friends to swap with dramatically reduces the cost of filling an album. In fact, it will almost definitely pay off in future swaps if you go out right now and buy starter packs for all your friends...

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2014-05-26 22:42:00

## 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.

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Comments in green were written by me. Comments in blue were not written by me.