No two card shuffles will EVER be the same.
 

That doesn’t sound right, does it? In all of human history, and all of human history to come, no matter how many times a deck of cards is shuffled, the same combination will never, ever, ever come up. How can that be? Well…

The 52 cards in a deck can be arranged in...
8,065,817,517,094,390,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
...different ways.

What if we were to put every person alive on the Earth, right now, on this shuffle project? According to the U.S. Census Bureau, the total population of the Earth, as of 02/04/08 at 15:40 GMT (EST+5) is 6,648,429,413.

So, each person would have to shuffle...
1,213,191,419,513,742,831,701,174,895,214,300,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
...decks of cards before all the possible shuffles were exhausted. What if we gave them a little time to work on that? How about 100 years…

If every person alive on the Earth, right now, shuffled continuously for the next hundred years, they would have to shuffle...
3,847,004,754,926,886,198,950,960,474,423,900,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000
...decks of cards, every second, before all the possible shuffles were exhausted.

We don’t seem to be getting any closer to achieving all those shuffle combinations. How long would it really take? Let’s put those 6,648,429,413 people on planet Earth to work, on a more realistic goal: shuffling one deck of cards every 10 seconds, 8 hours a day, 5 days a week, and see how long it takes.

360 shuffles per hour
2,880 shuffles per day
14,400 shuffles per week
5,256,000 shuffles per year
times 6,648,429,413 people is 34,944,144,994,728,000 shuffles per year!

We should see all those shuffle combinations coming up in about...
1,534,592,373,876,406,012,176,560,121,765,600,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
...years. Don’t hold your breath.

How do you derive this number? The number of permutations is 52! (= 1 x 2 x 3 x 4 x . . . . x 51 x 52), which is 8.0658 x 10^67. Your number seems to be 8.0658 x 1000^67, which makes a slight difference.

Does this work for Old Maid?

Flint is crazy.

I took 8.06581751709439 e+67, copy/pasted 6.7 lines of ten groups of ",000" then put commas in the original figure, like this "8,065,817,517,094,39" and then pasted these on top of the zeros so that it was the same length. Oh, I left the "8," on a serarate line, because it was before the decimal.

What did I do wrong? Oh. I did 67 groups of ",000" instead of just zeros.

Nothing. You're crazy.

Yeah, so? I just wanted to show the zeroes, so you could see how obscenely gigantic the number is. 52! is not likely to be exhausted during the course of human history.

Of course, the part I didn't include is how these are not really random deck shuffles. Card are arranged in typical, repeating patterns, due to the fact that we are shuffling from a partially ordered state, based on the card combinations that happen during the course of card games.

Anybody care to take a crack at that?

Flint,
I respectfully disagree, but don't have time now to work through the math.

flint--i've a huge crush on you.


deal.

Combinations of card shuffles, 52! is 8.06581751709439e+67.

Population of the Earth (possible number of people that could shuffle a deck of cards) is 6,648,429,413. That’s 1.2131914195137428317011748952143e+58 shuffles per person.

Divide 1.2131914195137428317011748952143e+58 shuffles by number of seconds per year (31536000) and you get 3.8470047549268861989509604744239e+50.

That means: if every person alive on the Earth started shuffling one deck of cards per second, it would take 3.8470047549268861989509604744239e+50 years to get through all the combinations.

Deal? There's 8.06581751709439e+67 different ways I could take that.

Flint, here's what I'm thinking:
There are 52! (52 factorial) ways that the first shuffle can come out.

The probability of the second shuffle exactly matching the first is 1/52!.

The third shuffle has a 1/52! chance of matching the first shuffle, and a 1/52! chance of matching the second shuffle, so by the third shuffle, there's a 3/52! chance of two shuffles being the same.

The 4th shuffle has a 1/52! chance of matching the first shuffle, and a 1/52! chance of matching the second shuffle, 1/52! chance of matching the 3rd shuffle, so by the 4th shuffle, there's a 7/52! chance of two shuffles being the same.

Etc...

The question should be, "how many shuffles would be required to have an X % chance of two shuffles matching?"

That's interesting.. I'd never really thought about it. (And believe me, the question of card arrangements is not so esoteric to blackjack and poker players, among others.)

EDIT: HLJ, Flint was being sensationalistic in his thread title. I think his point (though with Flint it can be hard to tell) is that there are so many combinations it would take "forever" to go through them all, even guaranteed that there would be no repeats.

Flint being sensationalistic ?!

eee gadds!

say it isn't so!

Heavens to Betsy!

[/great big globs of sarcasm]

Of course it's sensationalistic. I want someone to prove me wrong.

What are the odds of one particular permutation, of 52! possible permutations, repeating? It's been a few years since I took statistics, but I think this should be pretty easily answerable. With the right (TI-83/84) calculator.

What I'm saying, based on what I can see (and this is very counter-intuitive) is that in the history of this planet, we can be almost %100 certain that the same card shuffle has never come up, and never will. Of course, it could happen five minutes later, but the probability of this would be so vanishlingly small that it would be considered impossible for all practical purposes.