Mathematical equation for cutting cost of album

If you thought filling a World Cup sticker album was all about fun, think again...

As Russia 2018 approaches, the thoughts of millions of schoolboys and girls turn to squeezing more pocket money out of their parents to complete their World Cup sticker albums.

But mums and dads – or at least those of an academic bent – can rest a little easier in the knowledge that a mathematical model has now been developed for the cheapest way yet devised to complete their collection.

Sammy Ford, a consulting actuary with a degree in mathematics from Balliol College, Oxford, has calculated that filling a Panini album could cost as little as £161.65 — around a fifth of previous estimates.

She did this by building a model that assumes collectors swapping with friends and buying the last 50 stickers needed to complete the 682-strong set directly from the company.

She said: “Armed with this information, those determined to fill their albums should be able to do so in the most cost-efficient way possible.

“I’m sorry to anyone who feels this level of mathematical analysis takes the fun out of it but I’m sure many parents – and adult collectors – will be more interested in how much money they can save as a result.”

Sammy originally had the idea of working out the cost as a potential test for graduate applicants to her actuarial consultancy company APR.

She was encouraged by the company, which has several Panini enthusiasts, to develop the idea further.

A total of 682 stickers are needed to fill the book.

In theory collectors would only have to buy 137 packets of stickers, at a total cost of £109.60, to complete the collection, but this assumes they never get any duplicates.

Sammy says a simple calculation shows there’s more chance of winning the UK national lottery than for this to happen.

She began with a basic set of assumptions: that stickers are sold in packets of five, costing 80p each, that no packet contains the same sticker twice, and that all stickers are equally likely to be in a single packet.

Using the mathematical equation – e_k =[1/(1−Pr(k,0)] ×(1+∑ i=1,5 Pr (k,i) ×e _k+i) - she was able to calculate that a single collector would expect to need to buy 966.4 packets of stickers at total a cost of £773.12.

Allowing for the fact that collectors can buy up to 50 stickers directly from Panini at a cost of 22p each, the average cost plummets to £294.20 for individual collectors assuming they order the last 50 stickers in this way rather than the first 50.

“This might seem counter-intuitive, given that 50 stickers represent far less than a tenth of the book and yet buying these directly saves over 60% on expected cost,” she said.

“This happens because the more stickers you have, the less and less likely it becomes that the next sticker will be new. The last sticker alone takes us an average of 136 packets to find. Cutting off that tail end of the search saves an enormous amount of money.”

But these numbers still assume that all collectors operate independently, rather than working together on the playground principle known as ‘the swapsie’.

Factoring swapsies into the equation makes the maths difficult and so it was to cover this scenario that Sammy developed her model.

Based on five friends swapping fairly to maximise each person’s chances, Sammy ran her model through 1,000 different simulations, bringing the average cost down to £329.40 even if you don’t order any individual stickers directly from Panini. Of course the more friends you swap with, the cheaper it gets.

Panini insists all stickers in the collection are printed in equal quantities and Sammy didn’t dispute this but there are plenty of claims online that the 50 national association and tournament badges – known to collectors as ‘shinies’ – are more difficult to find. As you’d expect, if this were true it would make the whole thing even more expensive.

Combining all those factors, assuming five collectors were sharing their swapsies, that the final 50 stickers were bought directly from Panini and even assuming the conspiracy theorists are right and that shinies have a relative probability of 0.5 compared with the rest of the stickers, Sammy ran 2,000 further simulations to arrive at a final average cost of £161.65.

A spreadsheet of Sammy’s complete workings is available at https://aprllp. com/just-how-expensive-is-the-paniniworld-cup-album/.