How Many Shuffles Make a Deck Random?

Every card game starts with a shuffle, and every card game is only fair if that shuffle really mixes the deck. So how many shuffles does it take? The famous answer is seven riffle shuffles, and the maths behind it is a lovely piece of probability. Here is what it means, why fewer is not enough, and why a casual overhand shuffle needs far more.

How Many Ways Can a Deck Be Ordered?

A 52-card deck can be put in 52 × 51 × 50 × … × 1 different orders. That number, written 52!, is about 8 followed by 67 zeros. Even if every person on Earth had shuffled a deck once a second since the universe began, they would have seen only a vanishingly small fraction of the possible orders. So a truly well shuffled deck is almost certainly in an order that has never existed before.

The Seven Shuffle Answer

In 1992 the mathematicians Dave Bayer and Persi Diaconis worked out exactly how close a deck gets to perfectly random after a given number of riffle shuffles, the kind where you split the deck in two and let the halves fall together. Their model measures the distance from a perfectly random deck, from 100% (badly mixed) down to 0% (perfect). These are the values from their formula, which we recomputed:

Up to four shuffles the deck is still far from random; it only starts to mix at five. Then the distance roughly halves with every extra shuffle. Seven is where it first gets reasonably close, which is why seven became the rule of thumb. If you want to be very sure, nine or ten is better.

Why Fewer Shuffles Leave Patterns

One riffle shuffle splits the deck into two packets and interleaves them, so the cards from each packet stay in their original order. After one shuffle the deck is made of at most two rising runs of the old order, after two shuffles at most four, after three at most eight. A clever player (or a magician) can use those runs. Only after about five shuffles are there enough runs for the old order to start disappearing.

What About the Overhand Shuffle?

The overhand shuffle, where you slide small packets from one hand to the other, mixes the deck much more slowly, because cards mostly move in blocks and close neighbours tend to stay close. Mathematical estimates put the number needed for a well mixed deck in the thousands, far more than anyone does at a kitchen table. If you shuffle overhand, add a few riffles or a good smoosh, where you spread the cards face down on the table and swirl them around for a while.

Does It Matter at Home?

For a simple game like War, a lazy shuffle only means the same cards keep meeting. In games where you guess or plan, like Higher or Lower, a poorly mixed deck can make the next card more predictable than it should be. Card games are only as fair as their shuffle, just as people are not good at picking at random: humans are bad at being random.

Computers shuffle differently. A program can produce a perfectly mixed order in one go, by picking the cards one at a time at random (the Fisher–Yates shuffle). That is only as good as its source of randomness; see how random number generators work.

FAQ

How many times should you shuffle a deck of cards?

About seven riffle shuffles, according to the 1992 work of Bayer and Diaconis. Fewer leave clear patterns; nine or ten get even closer to perfectly random.

How many possible orders does a deck of cards have?

52 factorial, about 8 followed by 67 zeros. A well shuffled deck is almost certainly in an order that has never existed before.

Is the overhand shuffle random?

Only after a very large number of shuffles, estimated in the thousands, because cards move in blocks. Add riffle shuffles or a smoosh to mix properly.

Why is one riffle shuffle not enough?

One shuffle only interleaves two packets, and each packet keeps its old order, so the deck is still mostly predictable.

How do computers shuffle cards?

A good program uses the Fisher-Yates method, choosing each position at random from the cards left, with a good random number generator.