Have you ever been dealt the same hand twice in a row when playing a game of cards? Maybe that deck wasn’t shuffled as well as you thought. How many times should you shuffle?
How many shuffles does it take to randomize a deck of cards?
The answer, of course, depends on what kind of shuffle you consider. Two popular kinds of shuffles are the random riffle shuffle and the overhand shuffle. The random riffle shuffle is modelled by cutting the deck binomially and dropping cards one-by-one from either half of the deck with probability proportional to the current sizes of the deck halves.
In 1992, Bayer and Diaconis showed that after seven random riffle shuffles of a deck of 52 cards, every configuration is nearly equally likely. Shuffling more than this does not significantly increase the “randomness”; shuffle less than this and the deck is “far” from random.
In fact, it is possible to show that five shuffles are not enough to bring about the reversal of a deck. So it is somewhat surprising that just two shuffles later, every configuration is possible and nearly equally likely.
By the way, the overhand shuffle is a really bad way to mix cards: it takes about 2500 overhand shuffles to randomize a deck of 52 cards!
Story Credit: math.hmc.edu
Photo Credit: Max Pixel