Expected # of Purchases to Complete the Panini World Cup Collection

The Panini World Cup sticker collection is popular every 4 years as fans try to collect all players, stadiums, teams, and more. This year's edition includes 682 stickers to collect, which prompts the question: just how many sticker packs do you probably need to buy in order to complete the set?

The initial book comes with 26 stickers and costs $2. After that, every pack of stickers comes with 5 each and costs $1. I make the following assumptions:

• You don't trade any duplicates with other collectors, so the only avenue you have to get new cards is to buy more
• You don't buy more than one book
• The randomness of cards follows the discrete uniform distribution (every card is equally likely)

I ran 10,000 simulations to determine the expected number of purchases you would need to make to finish the book. In the median (most likely) case, you need 4,707 stickers to guarantee no duplicates. That's 1 book and 937 ticket packs, or $939 in cost.

Some additional metrics from the simulations:

• Minimum to complete the book ("luckiest" trial): 2,986 stickers/1 book + 592 packs/$594
• Maximum to complete the book ("unluckiest" trial): 10,513 stickers/1 book + 2,098 packs/$2,100
• Average to complete the book (expected value, skewed by the extremes): 4,852 stickers/1 book + 966 packs/$968

The distribution is skewed right (see below), and since I'm using simulation, you can read the below percentages as "X% of the time I will complete the book in Y packs or less":