How Many Shuffles Does it Take to Fully Randomize a Deck of Cards?

Описание: How many shuffles does it take to fully randomize a deck of cards?

The answer to this question fully depends on what type of shuffle you are using. Most people immediately think of the “Riffle” Shuffle, which is when the deck is divided into roughly equal halves, and brought together.

With a bit of math, we can determine that a brand-new deck will be effectively randomized after seven riffle shuffles, assuming that the cards are in “New Deck Order”. A paper written by Brad Mann of Harvard University’s Department of Mathematics explains why. It’s a bit complex, but basically a shingle riffle shuffle won’t result in a totally random deck, since many of the cards will be in a predictable decision. The top card will likely remain in the top position, and while it might be in the second or third position, that’s not really random.

However, every additional shuffle increases the likelihood that a given card will be in any given position. After two shuffles, you can say with confidence that the top card is near the top, but you can’t necessarily declare that it’s in a specific position. If we were to keep shuffling, we would increase this randomness.

To call the deck truly “random,” every possible combination of cards needs to be equally likely, and that occurs after seven riffle shuffles. Six shuffles is much less random - but eight shuffles won’t make the deck significantly more random. Seven shuffles should do the trick in a real-world setting.

Of course, there are more than just one way to shuffle a deck of cards. If you prefer something like the Overhand Shuffle, then make sure you’re ready for a workout, because you’ll need about 2,500 shuffles to get that same level of randomness, as this method has only a slight effect on the order of the cards.

There’s also the shuffle that’s a favorite of card magicians out there: the Perfect Shuffle. This one requires cutting the deck into equal, 26-card piles and then perfectly interlacing them, hence the name. If the top card stays on top, that’s referred to as an out-shuffle. If the top card moves to the second position, it’s known as an in-shuffle.

Although the Perfect Shuffle sounds like it might be a better option than the Riffle for randomizing a deck, an out-shuffle will return the deck back to its original order after just 8 shuffles. Similarly, the in-shuffle will do the same after 52 implementations. So while it’s certainly perfect for arranging the cards as you need, it’s not so useful if you’re planning on playing a friendly game.

With all that said, if you riffle shuffle seven times, you can count on a high degree of randomness. But that’s sort of an understatement, there are actually more ways to arrange the order of a deck of cards than there are atoms on Earth. After your seven shuffles, all of those arrangements are about equally as likely.