# The World Cup album: how much does it cost to complete the most controversial sticker collection in history?

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## Completing the World Cup sticker album, of 670 stickers, according to a mathematical model costs €940 with a standard deviation of €170

Messi, Cristiano Ronaldo, Mbappe, Modric, Luis Suarez, Benzema, Neymar or Lewandowski. The list of emblematic players appearing in this World Cup sticker album is one of the most dazzling in history. The presence of so many stars has attracted millions of collectors from all corners of the planet and has even become a matter of state concern. In Argentina, the frenzy unleashed to get Messi’s sticker has been so great that it has led to a shortage, and the government has had to mediate between the production company and the sellers to stop the shortage and speculation.

Although getting the Messi sticker with a bit of luck could cost €1 (the price of a pack), there has been much speculation: for the most unlucky, the price will reach €90.

The goal is to get 670 different stickers to complete the collection. Taking into account that each envelope contains 5 stickers, the minimum price to complete it is €134 if the impossible happens that the collector gets several stickers in all envelopes. But without the black market, trading or speculation, our calculations conclude that one would realistically have to invest €940 with a standard deviation of €170 to get the full collection of World Cup stickers. The album comes out for about the same price as a simple electric bicycle. Let’s see how we get to this result.

At the beginning of this type of collection, almost all the stickers that we find in each envelope are new. Soon repeated cards, the repes, appear. Where any collector would find frustration, we scientists see an opportunity to do an experiment. In this way I added paper and pencil to my World Cup collection, noting how many new stickers I obtained for each envelope purchased. While this measure is interesting, I was missing something else to know if my luck was really good or not. To do this, I decided to make a mathematical model of what happened.

Mathematical models are very useful for scientists because in reality we often only have an observation of the facts. However, given various random conditions, models can give us many observations, increasing the available information.

For example, the curves describing the number of people infected with Covid-19 as a function of time were based on a single observation. However, those curves include the consequences of many arbitrary causes. To better understand the situation and decide what measures to take, the mathematical models gave many curves of the same phenomenon and even reported the probability of the different scenarios for the evolution of the pandemic.

The model of this collection is very simple. Our computer simulates 5,000 families buying World Cup stickers. Each family will find five random stickers in each pack and will continue to purchase packs until the collection is complete.

The only condition is that no two identical cards can be in the same pack. We obtain 5,000 curves (one for each family) representing the number of new trading cards as a function of the number of packs purchased. With all these curves, we can obtain the trajectory of the median (curve that has half of the most fortunate families above and half below at each point) and the range between the 5th and 95th percentiles.

The area between those percentiles gives us a lot of information as we need very good luck to be above it (5% better) or very bad to be the opposite. This means that the vast majority of routes will be included in this area. As we can see in the figure, my collection has always been in the range between the 5th and 95th percentile. Plus, I was lucky because the deviations from the median were above it.

The model shows us that my experience brought happiness. However, this model can tell us more, such as how much we need to complete the collection.

By taking the average of the number of envelopes each family had to buy to get the 670 stickers, we arrive at an investment of €940 with a standard deviation of €170. This result indicates the importance of luck in completing the the collection, as it can change the final price by hundreds of euros. By the way, similar models are used in ecology to create dilution curves, which show the number of observed species as a function of the number of samples. Under the eye of a statistician, a biologist looking for different species in an ecosystem is not so different from a collector hunting for new maps.

Finally, we modified our model to include the exchange of ‘repes’. Now our families are grouped into pairs exchanging cards. The exchange is fair, that is, your ‘repe’ of chrome in exchange for mine. The result is that the average price to complete the collection is €660 with a standard deviation of €150, a much lower price than in the case of an isolated family. In short, it is better to have a friend than to be lucky.

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