Random Facts About … the Randomness of a Shuffled Deck

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By David Martin –

A lot of crazy stuff has come out of our internet tubes but none crazier than this: If you properly shuffle a 52-card deck, the resulting arrangement of cards is likely never to have come up in the 700-year history of playing cards and will not ever come up again for as long as cards are shuffled. Each thoroughly shuffled deck of cards is, in all likelihood, unique, meaning it has never existed before and will never exist again. 

We didn’t believe it either. Mathematicians explain, however, that there are so many possible combinations that 52 playing cards can randomly make … that that number is so gi-normous (not an official mathematical term) that the chances of two exact combinations ever occurring are practically nil. The number of possible combinations that a 52-card, properly shuffled deck can make is: 80,658,175,170,943,878, 571,660,636,856,403,766,975,289,505, 440,883,277,824,000,000,000,000. 

To give an idea of how big that number is, one Reddit source explained it this way: Put 10 billion people on each of one billion planets in each of 200 billion solar systems in each of 500 billion galaxies. Put all those people to shuffling cards at a million shuffles per second. Have that shuffling going on since the universe began 14 billion years ago. STILL all the possible combinations of a deck of cards would not have been shuffled. 

We refer to a “properly” shuffled deck of cards. A riffle shuffle is the most common type of shuffle: the deck is divided into roughly half, the edges of the two halves are put together, and then those edges are rapidly raised and lowered (riffled) in a way that puts cards from each half into the other half. Mathematicians have figured out that seven proper riffle shuffles will result in a randomly distributed deck of cards. The overhand shuffle is when clumps of cards are pulled from the deck and thrown back in randomly. Don’t trust this shuffle. It takes 10,000 overhand shuffles to properly randomize a deck of cards. In a casino, you might have seen dealers shuffle cards by putting them facedown on the table and then moving them all around. This moving cards around on a table has to be done for a full minute to achieve a properly shuffled deck. 

Here’s a card trick you can perform for (patient) family and friends. Riffle shuffle a deck of cards seven times, study how the cards are arranged, and then act surprised. Put the deck in front of your audience and say, “I’ve checked the order these cards are in and I’ve determined that no deck of cards has ever been shuffled to achieve this particular order. Never. In all the centuries cards have been used, this order has never occurred before. What’s more, if people keep shuffling cards for billions of years from now, a deck of cards will never be produced in this combination ever again. This truly is a unique deck of cards.” 

You might even try to sell your one-of-a-kind deck of cards to a brother-in-law for twenty bucks. But by then some smart-aleck kid will look it up on the internet and you’ll be busted. “It’s just math,” the kid will say. 

Just math? Ha!

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