Mathematics of Card Shuffling: Probability and Predictability

Have you ever shuffled a deck of cards? Even if you only play online casino games, the chances are you probably have. But did you know it takes just seven riffles — a term you’ll understand shortly — to achieve a completely unique and random sequence? If you’re shuffling properly, that is.

After seven casino-style riffles, your deck bears an arrangement that’s almost certainly never existed before. Even if you shuffled for a thousand lifetimes, you’d likely never see that same card sequence again. Keep reading to learn more about this and other mind-blowing facts and the techniques used by casino dealers.

A Clear Definition of Shuffling

A shuffle is an attempt to randomize the order of the cards in a deck, usually for games or magic tricks.

When you or your friends shuffle for a home game, it probably wasn’t a true shuffle. Most people give the deck a few quick mixes, which is enough for a friendly round of Texas hold’em. However, a real shuffle is much more than that.

There are several types of card shuffles, and if you’ve been to a casino, you’ve probably seen all of them. Here are the four most common and important styles. 

• Overhand shuffle: This is your classic, amateur-style shuffle that everyone has performed at their home game. It was likely the first way you learned to shuffle a deck of cards. Hold the cards in your dominant hand and slowly transfer them to the other, mixing them up as you go.
• The riffle: A professional dealer most commonly chops the deck into two piles before quickly interlocking them. A card from one stack is laid after a card from the other repeatedly until the deck is back to one pile.
• Strip shuffle: Casino dealers often combine this with a series of riffles. The deck is held in the air and cut near the top before placing that portion onto the table. Another cut is performed, placing those cards on top of the first portion. This series could involve anywhere from three to eight running cuts.
• The wash, or mix: The deck is spread all over the table and mixed together, with no apparent technique. After spreading the cards around for some time, the dealer puts the deck back together.

Probability and Predictability of a Deck of Cards

Maybe you already knew about shuffling. But when it comes to a deck of cards, probability is something most people know much less about. As mentioned, it’s true that with just seven riffles, you’ll have a unique sequence. That’s a sequence that hasn’t been seen before and will never be seen again. 

Surprising, right? But the math becomes stranger. Imagine you started shuffling at the beginning of the observable universe over 13 billion years ago. If you completed a seven-riffle shuffle every second, you’d likely never have seen the same deck of cards order twice. 

For some context, the total number of combinations you can make is 8.0658 x 1067. But the universe is only around 1 x 1018 seconds old. After a true shuffle, you will more likely become the President of the United States or be struck by lightning than find two decks in the same order.

The Probability of the Riffle Shuffle

After one riffle shuffle, you could guess about 30 cards correctly, in theory. After two riffles, that number drops to 19, followed by 13 on the third. With each shuffle, the probability of correctly guessing the cards drops. 

After seven riffles, a deck of 52 cards should be absolutely random, which makes the outcomes unpredictable and fair. 

Lowering the predictability to the absolute minimum is vital for game operators and players. The riffle shuffle is the best way to randomize the order of cards in a deck, which is why it’s a trusted method. But at the same time, it looks incredibly cool, too.

Card Shuffling at Live Casinos

Shuffling was always traditionally performed by hand. It still is in some places, but with the evolution of playing cards and casino games, you’re more likely to see automation. When you walk into a modern casino for a game of blackjack, you may notice small black boxes set into the tables.

These are automated shuffling machines that perfectly blend decks of cards, usually more than one at a time. Instead of a dealer practicing their shuffling techniques for hours and hours, these machines offer more reliability. Pre-shuffled decks speed up the flow of a game, too.

For live poker, shuffling machines are recessed, sitting flush with the tabletop. Blackjack shuffling machines used by a live dealer online will either sit on top of the table or somewhere off-camera. 

How Do Online Casinos and Poker Sites Shuffle Cards?

You might know how many ways to shuffle a deck of cards there are, but what about online? How do casinos and poker rooms make sure their decks are perfectly shuffled?

Online gaming companies use two main shuffling methods:

• True RNG: A system that uses a genuinely unpredictable means of generating numbers.
• Pseudo-RNG: These generate numbers with a computer algorithm that seems random but isn’t truly.

Whichever random number generation system is used, they are always highly regulated, certified, and monitored. While you may use math skills in gambling games, RNG companies use them to guarantee fairness.

Can Players Use Card Shuffling To Cheat?

Cheating was rampant in the early days of gambling, but security has tightened a lot since then. Modern techniques, like surveillance and shuffling machines, all make it harder to cheat. 

To cheat via shuffling, someone must either pre-arrange the deck or predict the eventual sequence. Since the shuffling machines have become a staple in casinos, neither is an option. Even for a dealer handling the cards, arranging the deck is almost impossible. 

Online attempts to cheat don’t usually exploit shuffling systems because of their complexity. Hackers typically try to view the cards of other players instead. Luckily for players, security is a top priority for all gaming companies. The reputation of every major online or offline game operator depends on it.

Shuffling Is No Game

Card shuffling math is intriguing, but you need to know about mathematics and gambling strategies for your favorite games, too. Learning more math can certainly make you a stronger poker player, for instance. Even sports bettors should be studying odds and probability to make smarter wagers.

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