Card Shuffling Will Blow Your Mind

We’re continuing our discussion of things Dr. Su’s visit made us think about.

In our previous post, we learned that if you do the Riffle Shuffle, then you should do it 7+ times to get close to perfectly mixing the deck.  Remember, by perfectly mixed we said that all 52! configurations are equally likely.   We’d like to revisit that statement.  First, let us remind you that

52! = 52 \times 51 \times 50 \times 49 \times \cdots \times 4 \times 4 \times 3 \times 2 \times 1

Why is that the number of configurations?  Well, for each configuration there is 52 possibilities for the first card, and once that’s been chosen, there is 51 possible cards for the second card, and so on.  To get the total possible number of configurations you have to multiply all these possibilities together.

Okay, but how big of a number is 52!?  Pretty big, you say.   For this and all other calculations in this post we’ll use Wolfram Alpha.  That’s a search engine recently released by the makers of Mathematica which is especially good at working with numbers.  In particular, it can do most of the things Mathematica can do (like integrals! Try typing “integrate sin(x)” into Alpha.).

Using Alpha, we get that 52! is (after rounding down) around 8 \times 10^{67}.  That is, 8 with 67 zeros after it.  Okay, that sounds pretty big.  To give you some idea, there is an estimated 1 \times 10^{80} atoms in the universe.  So it’s up in that range.

Here is the amazing thing.   Let’s estimate (actually we’ll way overestimate) how many different configurations of the standard 52 card deck have occurred since shuffling was invented.   Here is the numbers we will use (hopefully you’ll agree that they are all reasonable or, more likely, overestimates):

  • Let’s say playing cards have been around for 10,000 years (actually, it’s more like 1,500 years).
  • Let’s say each of those years has had  100,000,000 seconds (actually 365*24*60*60 = 31,536,000, but there was leap days, etc., so we’ll round way up).
  • Let’s say during each of those seconds there was 10,000,000,000 people living on the Earth (actually right now there is around 6,800,000,000, and that’s the most it’s ever been).
  • Let’s say that for every second of every one of those 10,000 years all those people were doing one riffle shuffle a second.
  • Let’s also assume that every single shuffle resulted in a brand new configuration that never occurred before.

Hopefully you agree that every assumption is a ridiculous overestimate.  Multiplying it all together we see that there has been a total of

100,000,000,000,000,000,000,000,000

That’s a huge number!  Or is it?  In scientific notation, that is 1 \times 10^{26} different configurations which have happend.  Said a different way, as a fraction of the total possible only

\frac{1 \times 10 ^{26}}{52!} \sim 1/10^{42}

of the total number of configurations have occurred.

Said a different way, if you have shuffled your deck a good 10+ times so that is well mixed, then the configuration your deck is in has a one in 10^{42} chance that it has occurred before.  In comparison, if you buy one Powerball lottery ticket, your odds of winning is one in 195,249,054.

That is, you are more likely to win the next 5 Powerball drawings then to have a configuration which has occurred before!

The odds are staggeringly high that your well mixed deck of cards are in a configuration which has never, ever occurred before in the history of cards!

If that doesn’t blow your mind, then you are probably better off playing cards with these guys:

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