How many packs to complete the album?

The World Cup was recently over. Along with the competition, the sticker album also arises, it’s a quite big tradition, but I’ve never joined it. I got interested in the statistics behind it and asked myself how many stickers you must buy to fill the album completely.

My approach is a statistical simulation, modeling each package, until the album is complete. The same procedure is repeated for a large number of runs to get an estimated distribution of the total number of packages/stickers that are necessary to complete the album. First, I tested the convergence of the routine, initially based on 2 unanimous assumptions: the distribution of the stickers is uniform (that means, you have an equal chance to get any of the stickers) and that there are no repeated stickers for each pack (this one is maintained for all the tests here). Secondly, I tested what are the advantages of buying the missing stickers (from 1 to 50). Finally, two cases where the distribution is not uniform are evaluated: for a selected nation, the stickers are more abundant (from +10% to +50%) or rarer (from -10% to -50%) than the others.

This analysis can be performed for any album, being the number of stickers in the album and the number of stickers in a pack the necessary variables. So, for this case, the values for the Panini World Cup sticker book are selected:

  • 681 stickers in the album;
  • 5 stickers per pack.

Also, the possibility to buy missing stickers directly from them (maximum of 50) is also considered in this work.

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