## Friday, April 13, 2012

### Completing a Panini Sticker Album

When I was a kid my main ambition was to fill out the Panini Italia 90 sticker album. I promised myself when I was old and rich I would buy all the stickers I needed to fill out a championship sticker album. Some kids aim to play for Ireland, I aimed low. Turns out I am not rich but I am nerdy so at least I can work out how much it would cost to fill out the album.

This is called the Coupon Collector Problem.
"Given n coupons, how many coupons do you expect you would need to draw with replacement before having drawn each coupon at least once?"

There are 539 stickers in an album to collect. The first sticker you buy is going to be one you need, a 539/539 chance of getting one you need. The second has a 538/539 chance of being one you do not have as there is one it can clash with. This keeps going until for the last sticker every new sticker has a 1/539 of beng the right one. The formula for the number of attempts you would need to calculate the coupon collector number is 539*Harmonic Number of 539. Which according to Wolfram Alpha is 6.867858*539=3701.77. The stickers are sold in packs of five. Which means 740 packs.

On amazon a box of 100 packs cost £43.95. This would mean (assuming you could get .4 of a box at that same price) that filling the album would be expected to cost 325.23 pounds.

Panini sell the stickers at 14p each. So to buy the stickers individually would cost 539*.14=75.46 pounds.

Is there some kind of mixture of boxes of random stickers and individual ones that means you can fill in the album for as cheap as possible. Say the individual stickers can be bought at the price of a box. So thats 0.1162 pounds for each sticker. Which is not that much of a saving. It becomes more efficient to buy individual stickers than random packs at this price after about 90 stickers.

The last sticker takes on average 539 stickers to be bought to find it. So this last sticker costs 0.1162*539=62.63 pounds if you buy it in packets not individually.

But are the tickets independent? As in are some rarer than others and is each pack random? Some people studied this in the very cool paper 'Paninimania: sticker rarity and cost-ef fective strategy'
"We consider some issues related to the famous Panini stickers devoted to the football world cup. In particular, we address the following questions: is there a planned shortage of some stickers? What is a good cost-e ffective strategy to fill in an album?"
Which proves amongst other things I am not the only nerd that still wants to fill in a Panini album.