Pokémon turns 30 this year. Beginning life as a video game which spawned a wildly successful trading card game and an anime series, the franchise – based around collecting an assortment of fanciful “pocket monsters” – has become an inescapable piece of the world’s pop-culture furniture.

A dedicated 30th-anniversary card set is due to be released to celebrate the occasion. A wave of nostalgia has created high demand for trading cards, with resale prices rising and sellers adopting anti-scalping measures and guarding against increasingly audacious thefts.

Behind the hype is a simple question familiar to generations of players: if you try to “catch ‘em all” by opening booster packs – sets of randomly allotted cards – how long would it actually take? And would you get bored before then?

Opening packs, virtually

To tackle these questions, we can turn to mathematics. I built and ran a computer simulation of six recent Pokémon Trading Card Game sets from the current Mega Evolution series: the base Mega Evolution set plus the expansion sets Phantasmal Flames, Ascended Heroes, Perfect Order, Chaos Rising, and Pitch Black.

The simulation model uses published “pull rates” – how often particular cards turn up – estimated from thousands of real booster packs opened by collectors.

I then created 10,000 virtual........

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