It depends on your method.
Pretty much every card game involves some sort of shuffling. Regardless of your method, the goal is always the same — ensuring that the cards are as randomized as possible.
In the Numberphile video below, Persi Diaconis from Stanford University explains how many times you need to shuffle using three common methods and the math behind the numbers he gives.
The first method he discusses is the “Riffle Shuffle,” also known as the “Dovetail Shuffle,” and it is the one pictured above. According to a theorem, seven of these shuffles are enough to get a properly mixed and randomized deck.
There are 1068 different possible arrangements for a deck of 52 cards, so investigating the likelihood of each using each shuffling method is not practical. Diaconis suggests a thought experiment.
If you shuffle a deck of cards and ask a friend to guess what the first card you turn over will be, his or her chance of guessing it right is 1/52 since there are 52 possible cards. If you do the same with the second card, his or her chances get better: 1/51. If he or she has a really good memory, the number of cards your friend is apt to get right is about 4.5 in the deck.
SEE ALSO: How to Win at Rock-Paper-Scissors, According to Math
It was determined that it takes seven riffle shuffles to properly mix the deck because fewer shuffles lead to a better chance of guessing more cards correctly, but more shuffling doesn’t change the probability.
His second, and least effective method, is the “Overhand Shuffle” where little piles of cards are dropped from one pile to another. This method has to be repeated ten thousand times to adequately shuffle the deck!
The third method he talks about he calls the “Smooshing Shuffle.” It involved spreading all the cards out in front of you and smooshing them around until they seem mixed. A dealer would need to do this for 30 seconds to a minute.
Diaconis goes on to explain the method behind finding out how likely a perfect shuffle is. Pretend that you have a deck of cards and look at the bottom card. In his case, it is a King of Hearts (KoH). Now, take the top card and place it anywhere in the deck. There is a 1/52 chance that it will be below the KoH. Now, take the second card and place it anywhere in the deck. There is a 2/52 chance that it will be below the KoH, and it is equally likely that it is above or below the card that you previously placed. If you continue in this manner, you will eventually have a perfectly randomized deck below your KoH and it will be your top card. If you finish by placing it randomly in the deck, the whole thing is random.
He calculates the likelihood of this happening and determines that it would take placing the top card somewhere in the deck 236 times:
52/1+52/2+52/3+52/4+...+52/52=236
Watch the video to see his explanation in action.
Editor's note (Jan. 5): Diaconis's calculations and our description of them have been edited for clarity.
Editor's note (Jan.6): Paragraph 9 has been edited so that the second card has a 2/52 chance of being under the KoH. The original article read "1/51 chance." The number of possible arrangements has been corrected to 1068. The original article stated 1068.
