The math of shuffling cards almost brought down an online poker empire

Summary

If you’ve ever shuffled a deck of playing cards, you’ve most likely created a unique deck. That is, you’re probably the only person who has ever arranged the cards in precisely that order. Although this claim sounds incredible, it’s a great illustration of how quickly large numbers can creep into everyday situations—with occasionally challenging consequences, as the developers of an online poker game painfully discovered in the late 1990s.The mathematics of card shuffling is quite easy to explain. To calculate how many arrangements 52 playing cards can have, you must go through all the possible shuffles. So logically, one of the 52 cards is placed on top, and once that’s determined, there are only 51 possibilities for the card below it. The next card has only 50 possible options, and so on. A 52-card deck can therefore be arranged in 52 × 51 × 50 × ... × 2 × 1 = 52! different ways.If you do the multiplication and round the answer, you will get a number with 67 zeros. That’s more than a quadrillion times as many ways to arrange these cards as there are atoms on Earth.On supporting science journalismIf you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.So clearly there are many, many different ways to arrange 52 playing cards. And yet to find out how likely it is that another person in the world will randomly create the same sequence of cards by shuffling, it’s not enough to simply calculate 1 / 52! That number simply indicates how likely it is to get one very specific shuffle of cards.There’s a more subtle question to consider: How likely would it be for any two or more people in the world to randomly shuffle a deck of cards in the same way?The Extraordinary Rarity of Each and Every ShuffleThis question is reminiscent of the birthday paradox. It works according to the same principle:...