Making History by Card Shuffling

Did you know that whenever you shuffle a deck of cards, it is quite likely that you are making history?

A deck of 52 cards can be ordered in 52! = 52 x 51 x 50 x…x 2 x 1 ways. This is because there are 52 ways to choose the first card, 51 ways to choose the 2nd, 50 ways to choose the 3rd, etc. But 52! is a very large number: larger than

8 x 1067.

How big is this number? Well, someone shuffling a deck of cards once per second since the beginning of the universe (believed to be about 14 billion years ago) would not have shuffled the deck more than 1018times.

Thus it is quite likely that any given configuration achieved through random shuffling has never appeared before in the history of shuffling!

Presentation Suggestions:
You might also compare 1067 to other large numbers or the number of stars in the universe (1023).

The Math Behind the Fact:
We ought to be a little careful here. In reality, there are many kinds of shuffles and not all of them involve randomness, e.g., see the Fun Fact Perfect Shuffles. In order for the result of a shuffle to produce an independent configuration, we must use a shuffle with inherent randomness.

But even a single random riffle shuffle does not make every configuration equally likely, so a single random riffle shuffle will not produce an independent ordering of cards. However a single random riffle shuffle produces nearly 252 possible configurations, which is still a very large number (4.5×1015), and after seven random riffle shuffles nearly every configuration is equally likely to occur, as explained in the Fun Fact Seven Shuffles. So it is still very likely that each random riffle shuffle is truly “making history”.

How to Cite this Page: 
Su, Francis E., et al. “Making History by Card Shuffling.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Lew Ludwig

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