no two decks of cards randomly shuffled have probably ever been in the same order

Here’s a fact that’s incredibly simple and very easy to prove, but I still find incredible.

Take a normal pack of cards—52 in total (not that the exact number matters)—and shuffle it.

A very simple process most people will have done at least a few times in their life.

Now take that randomly shuffled pack and lay it out in a line so you can see the sequence of cards.

Now just look at the cards for a second and the order they’re in.

You are the first person in all history to see a pack of cards in that order.

Never in the history of humanity has anyone ever held a pack of cards in that order.

Okay now to be fair we can’t technically prove this but it’s so overwhelmingly likely it’s ridiculous to deny.

How could this possibly be true?

A pack of 52 cards has exactly 52! possibilities, that is 52 factorial (52*51*50…*3*2*1)

Immediately you can see this is a pretty big number, but exactly how big you likely won’t have registered. That is approximately equal to:

80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

That is 8 with 67 zeroes.

Thus when you shuffle a deck randomly there is a 1/(8x10^67) chance that it is any specific combination. Again, to give an impression of how unlikely that is it is a 0.000000000000000000000000000000000000000000000000000000000000000000000125% chance.

That’s a bit of a mouthful so let’s only consider the first ten decimal places hereafter as if the chance is less than that it can be considered in effect zero.

If you shuffle a pack of cards 100 times the chance that any of these 100 are a specific combination is 0.000000000% to 10 decimal places.

If you shuffle it 1 million times it increases substantially to 0.000000000% to 10 decimal places… perhaps not substantially enough.

Let’s now try 10 billion: that’s if every human alive shuffles it once plus a couple billion more times. Now it’s far far higher at 0.000000000% to 10 decimal places… still not high enough.

Okay now let’s say that you shuffle a pack 1 trillion times? That is, dozens more than there have been humans in all history? Still 0.000000000% to 10 decimal places.

Now it’s already unlikely that in all history that collectively all packs of cards have been shuffled that many times; let’s go a bit (well actually quite a lot) higher to see if it still stands and shuffle cards 1 decillion times, that would be if every human ever alive shuffled over a trillion times each, now it comes out to *drum roll*… 0.000000000% to 10 decimal places.

And to clarify, no, it isn’t yet close.

If you consider it to 20 decimal places you still round to 0.0000000000000000000%.

So not only is it totally unfeasible that ever in all human history have two decks been randomly shuffled and come out the same, but we could multiply the number of humans by a billion, have them all shuffle a trillion decks each and it would still be a less than one in a billion chance that any would be the same as the one you just shuffled.

Wow.

I don’t know about you but to me that sounds pretty wrong; I can assure you however that it isn’t.

A few people in the comments have made the valid point that this is a misrepresentation of the situation as it takes a shuffle to be a purely random order when in fact it very much isn’t. The reason for this is often when people shuffle they are shuffling from an ordered pack and frankly don’t do a thorough job of it so the order is still not random, furthermore, people often use similar shuffling methods making it again more likely the order will come out the same.

This point is entirely true but doesn’t undermine the argument for the simple reason that the numbers are not close enough for it to make a difference. Exactly how much more likely than in the pure mathematical case it is in the practical case is almost impossible to gauge, but it could possibly be quite a great difference. As a result, if you calculated that there would in the pure case be a 1% that in all history the same order has been randomly shuffled to twice then it would be reasonable to reject this as not convincing as the reality could easily be ten times more likely at which point the chance is high enough you can’t say with confidence it hasn’t happened.

The issue is that the chance is so much lower than that. To take the last calculation, you could have every human ever alive shuffle a trillion decks each and the chance that any two were the same could still be increased by a factor of a trillion and still be 0.0000% to 5 decimal places.

In short, yes in the practical case the chances are more likely than represented enough, but the odds are so ridiculously small that this change is nowhere near enough to be relevant.