QI Card Shuffling - 52 Factorial 

So...you're saying there's a chance...

When Stephen started talking about the number and the stars and galaxies, all I could imagine was "Space is big. Really big. You just wouldn't believe how vastly, hugely, mind-bogglingly big it is."

imagine his face when he shuffles the exact same deck just a few hours later

I read two clever ways of putting it: Take out a digital countdown timer and set it to at 52! seconds. Scenario 1: -stand on the equator and take a step *once every billion years* -once you've circumnavigated the globe, take a drop of water from the Pacific -when the ocean is entirely drained, refill it and place a piece of paper down. Repeat *until the pile reaches the Sun* The first 3 digits haven't changed. You are 1/3000th done. OR Scenario 2: -shuffle a pack of cards and deal yourself five cards, *once every billion years* -when you get a Royal Flush, buy a lottery ticket -when you win the lottery, put a grain of sand in the Grand Canyon -when it is full, empty it out and take an ounce of rock off Mount Everest. Repeat *until the mountain is flat* The first 2 digits have not changed. You are 1/256th done.

If it seems impossible ask yourself this question, how many times has anyone inadvertently shuffled the deck to the original order?

Nice video. With all the certainty we can realistically have, his statement is correct (that it's the first time that order has appeared). And yet the amazing thing is that this proves that even though things may be statistically improbable, they are actually possible. Whatever the order of cards was that resulted from his shuffle, the odds of that order appearing were 1 in 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. And yet ... it happened.

Despite the odds, you'll still get yelled at by the guy next to you at the blackjack table for "taking his card" when you deviate from basic strategy.

Its still wrong to assume 100% certainty even with outstanding odds on your side

Those final zeroes are suspicious!

I wouldn't call it certain, but highly probable.

I took probability in College and I'm pretty sure I can't even grasp this in the nominal sense.

"Number of possibilities" "chances of a collision happening"

The way he described this seemed almost unbelievable to me, so I figured it out, and assuming 400 billion stars in the Milky Way (on the upper end of the average of current estimates), given the frequency of shuffles on all the decks on all the planets in the scenario described, it would take over 8.5 X 10^ 24 years (over 632 Brillion (corrected) times the age of the universe), to cycle through all possible permutations, unless there's an error in my figuring.

I just watched the Vsauce video which had this and they used 2 analogies one was imagine you had a timer with 52! seconds counting down and you are standing on the equator every 1,000,000,000 years take a step when you go around the world take a drop of water from the pacific ocean when you empty the ocean put a piece of paper on the ground and do this every time you empty the ocean until the pile of paper reaches the sun if you do this 1000 times you will be 1/3 of the way done or every 1,000,000,000 years deal yourself 5 cards every time you deal a royal flush buy a lottery ticket if the ticket wins the lottery throw a grain of sand into the grand canyon and do this every time you win the lottery once the ground canyon is full remove 1oz of rock from mount everest and do this every time the grand canyon is full and until mount everest is level once you've done that 256 times then the timer will hit 0

It's not fact it's just probabilities. You don't exhaust every possibility before you repeat stuff. All the stuff about "they would only now be done" would be assuming "every next shuffle will be a new order of cards until all possible orders have been exhausted." It's not an "absolute fact" it's just very very high probabilities.

People who understand math and probabilities know he is right. People who don't think he is wrong.

Its amazing because the large number comes from a relatively small number 52.Part of the fascination of cards

Stephen seems quite proud of himself.

So proud that something so simple can be so unique.... amazing if u ask me

To put it in perspective, there are about as many ways to arrange 52 cards as the number of grains of sand it would take to fill the whole Milky Way Galaxy

Stephen awarded himself points but I take them away for calling them a pack of cards when any decent person knows it’s a deck of cards.

* Some dude spits facts about math* People applaude. My type of show.

'I can say with all the mathematical certainty that is possible...' *Mathematicians all over the world cringing*

One way to help visualize this - imagine you place the cards in a specific order such as all aces together, all twos together, etc. and then imagine how many times you would have to shuffle the deck until they were in that exact same order again. It would absolutely never happen.

While what he says about 52! being a gigantic number and how long it would take for repeats and all that is technically true, it is only true if you are granting that every shuffle is, in fact different. The thing about random chance is that the probability of a repeat is never at 0% and it grows faster than the number of instances themselves (exponentially). For example when looking at birthdays there is a 50% chance that 2 people in any given group of 23 people share a birthday. When you have 70 people the probability is 99.9%. That is a crazy fact considering that there are 365 (70 days of the year is only 19% of the year) options and I'd imagine that a similar curve would be applied to card shuffling. Now while it is still a huge amount of time if the curve is anywhere near similar then we'd expect to see repeats about a fifth of the way through the time he said. That being said it's still highly unlikely that throughout human history we've reached anything near a fifth of all potential shuffles so he's more than likely still correct. Anyway forgive my nerdy nitpicking ramble, it is always cool to see 52! being explained.

For those new to QI, it's really not about maths at all. It's mostly just a fact show with lots of funny people. Highly recommended.

On your part. Probability that someone already did same shuffle is so small that you can call it absolute certainty.

Rain Man has verified this fact.

It's been a very long time since I've done maths beyond counting change, but if I'm correct, the chance of a deck of cards repeating itself is 1/52! * 100% ≈ 1.24*10^-66% ≈ 0.00000000000000000000000000000000000000000000000000000000000000000124%

If we're talking about statistics, then he should also have pointed out that a very small chance doesn't mean "never".

Mind. Blown....

This is not entirely true: shuffling by hand will never been a true random shuffling

I read a thing where it talked about 52! and how long that amount of Seconds would be. He described it like this: deal five cards (randomly), and wait for a billion years. Then, deal another five cards. When you get a royal flush, buy a lottery ticket. Keep doing the dealing and waiting thing untill you win the lottery. When you do, put a grain of sand in the Grand canyon. Keep doing this, untill you fill the entire Grand canyon with sand. When this is done, take one oz (ca 28 grams) of rock out of mount Everest. When it is level, all the time hasnt passed, you have to do this whole thing 256 times.

Casinos use pre-shuffled decks for baccarat. The casinos get them pre-shuffled from the manufacturer. In 2012, a baccarat table at the Golden Nugget casino in New Jersey had 2 consecutive decks that had been shuffled in exactly the same order.

Just to be sure, I think we should add a 53rd card soon to decrease those odds even more ;)

It is not impossible for that order to have happened before, however it is extremely unlikely.

It's not a mathematical certainty. Its a mathematical near certainty, because there is an infinitesimally small chance that it will have been in that order somewhere.

I'm pretty sure the birthday paradox applies here and the numbers are reduced by HUGE quantities.

It actually takes 7 riffle shuffles to fully randomize a deck of cards. If that was a new deck in order then the shuffles he did would not be sufficient to randomize them, although if they were randomized to start with then the order would indeed be unique

The comparison I heard was that there are more arrangements of 52 cards then atoms making up the earth. And the ratio between number of atoms of earth and arrangements of 52 cards is the same as $3.50 and the value of all money on earth.

And just think of all the things like this, that surround us in our daily lives, that we just take for granted that are even more complex than that

That's big, but not as big as 53!, or 54! come to think of it

That's some rain man information right there!

I think he is right as he says " THIS pack of cards has never been in that order"

scientifically speaking, we would assume he's correct because the chance of a repeat shuffle being 1.2e-66%

I believe this statistic whole heartedly imagine shuffling a shuffled deck of cards back to its original out of the packet in order state you basically couldn't do it it

I work in a casino and deal 150-200 hands of poker 5 days a week, so that's how many times I make history each day.

um, no. unless you have examined the result of every previous deck shuffle you cannot, with certainty, say that that shuffle is unique.

This would be great patter for a card trick...

this is 50/50 chance to get it

You can't really be certain the order fo a deck has never been in a certain order twice. You can only be certain that it would take a really long time to get all combinations.

so first the sun isn't there, and now he wants a nobel prize for shuffling cards! phil jupitus must be going bananas

Beautiful.

not strictly true. by sheer coincidence that particular combination could have been done already, though unlikely. also, in reality shuffling does not perfectly randomize the cards and decks are always packaged in order. so the order after the first shuffle of a fresh deck would be far more likely to match the order of other first shuffles than 52! to 1.

If you organize the enough amount of decks of cards in all the possible different ways and place them side by side they would correspond to a length of around 8*10^50 light years.

2:39 "I'm not impressed" face to cover up the fact she doesn't get it.

Mind BLOWN!

In an infinite universe, this exact same shuffle has been repeated an infinite amount of times on an infinite number of airings of this particular episode of QI and so too has this edited comment.

Well, fuck my old boots. You learn something new everyday.

Surely though as the cards are always in the same order when new out of the box, the probability getting all the cards in the same order after the initial shuffle (depending on how well the deck is shuffled) will be lower than if the cards were completely random before shuffling? Therefore, certain combinations of cards must be more common than others. Or am I just delusional?

I am an agnostic believer on this, I dont onow that the pack has never existed before but given the probabilities I think it so unlikely that i can say i do believe it has not.

You can't say that with absolute certainty based on probability. You can only say that it's VERY unlikely. It's possible that someone did an identical shuffle a day earlier.

i remeber watching this and thinking that has to be the greatest fact ever! Just incredible

52! is roughly 80 unvigintillion. Of course most people don't know how much a unvigintillion

What we're taught in medicine - never say never, and never say always. Of course there is a tiny chance that deck-order has existed before, a very tiny chance.

It's more like a 99.999% chance that it had not been shuffled before

Yup, the number is larger than the amount of atoms in our galaxy!... Unreal!

It is possible to repeat sooner rather than later because anything is possible at any given time. It literally can happen. This makes out that there is some counter and when it reaches max, suddenly it starts repeating the order of the deck. Sure a lot of combinations will have been played, but not all of them.

I’ve actually came to this conclusion myself a long time ago and have thought about it many times. And to convince myself I consider all the number of decks of cards in which all of one color is on top and all the other is on bottom plus all the decks that alternate between the two colors (26 factorial times 4-a huge number). Think of how unusual it would be to shuffle a deck of cards and have them turn out in either of those ways. All decks are just as unlikely. Their all equally unlikely.

No, it's not an absolute certainty, but it is a reasonable certainty. Like saying if you pick a person at random out of every person on the Earth 1,000 times, it's possible you'll get the same person every time, but it's reasonably certain you won't.

Watch vsauce explaining 52! Mind blowing

there is more chance of me shuffling a pack in exactly the same order as stephen than there is of Roma getting to the champions league final

There are problems with this, starting with his opening statement. It's impossible for anyone to say it's never been done since the beginning of time. (He later clears that up, saying that it is possible but mathematically unlikely.) A greater problem is with how he illustrates the unlikeliness of it. Try telling a friend what he said two days after watching this. ("Wait, was that shuffling a thousand times a second or a thousand times a minute?") A description I've seen said that gives a better idea of the magnitude of the number of possible combinations was simply that the number is greater than the number of atoms making up Earth.

Ah yeah, well, joke's on him. I've seen this one before

@IUPLOADALLSORTS: I'm sorry, but bigybop is right. It's not a fact. It's incredibly unlikely, but there's no way to prove that any individual shuffle has never been done before. Not only that, but the more new combinations that have been tried ADDS to the probability of the next one being a repeat.

Him presenting this has never been done before, and there's never been another him in the past. So naturally every moment there has never been done before. Though, with the cards, his math assumes that every order is equally likely. I doubt his shuffling was sufficient to make every order equally likely.

I think you're not grasping how big this number is. It's like the lottery, sure everyone can win, but did you? The chances to win (in the uk) are only 1 in 13 million... when the chances of shuffling a deck in the same order are 1 in ... 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000 millions ... Thing is, there have still not been enough decks shuffled on earth since the beginning of time for you to safely assume that the deck you shuffled is a first timer :)

Even that would be vastly more likely the odds "just" being about one in 7 short trillions as opposed to literally one in trillions of quadrillions. Thick people on here claiming this could have easily been done before don't realise this number is so unfathomably huge it's most easily comparable to the number of ALL the atoms of the known universe. Mind boggling isn't even the right word for it.

Did you notice before the clap at 2:30 he said THIS pack of cards has never seen that order before? Yes, because you broke them out the original packaging merely minutes earlier probably. So that deck of cards is only a couple of days old

So how did evolution create a simple protean that is more like a billion factoid?

The number of possibilities is less than an infinity, that means the chance is bigger than zero. So there is a chance... BTW this isn't nothing exceptional just basic math.

It seems quite extraordinary but for anyone trying to get their head around it just imagine it with a smaller number and scale it up. Lets say a deck has 2 cards. There are only 2 combinations (1,2 and 2,1) if it had 3 cards (3,2,1 3,1,2 2,1,3 2,3,1 1,2,3 1,3,2) it all scales up. A deck of cards with 2 cards has two possibilities (2! = 2x1) same with 3! (3x2x1 = 6) etc. it is amazing though that such a large number can come from just 52 cards

How many different ways can uno cards be arranged

I mean, someone could have. It's just very unlikely that someone has actually got the same deck, but it's POSSIBLE.

I don't think enough trillions fit into that number displayed there.

surely there are other variables to this because people shuffle the same way especially if you are doing a weave shuffle

Is that Gordon Gordon Wyatt

Not impossible, just very, very improbable. So, no, you can't know for sure.

The number of perfectly randomising shuffles to get a deck the same as Stephen's is indeed on the order of 52!. Specifically after 52! shuffles the chance of a deck the same as Stephen's having happened is 1-exp(-1). However the chance of any two decks being the same is far higher. To have a decent chance of any two decks matching you need on the order of the square root of 52! shuffles. 1.057x10^34 would give a roughly 50-50 chance. With trillions of people on trillions of planets doing 1000 shuffles a second you'd repeat a shuffle after about 4 months.

tl;dr Just nitpicking about probabilities. The number of attempts necessary to have a good chance of finding the same combination twice or more is way smaller than 52!. Seems to me that there is a misunderstanding of the probability to draw the same combination cards 2+ times. First of all there is a non-zero probability to draw the same combination twice in a row. 52! is simply the upper limit after which you are bound to repeat combinations. There is a well known problem that is similar to this: how many students do you need to have in a classroom for the odds of having 2+ students sharing the same birthday to be over 50%? Here the upper bound is 366, meaning that if a classroom has 366 students, 2 or more are bound to share the same birthday. Well for the probably of a shared bday to be >50%, there only needs to be 23 students in the classroom, assuming that bdays are randomly distributed. That's an order of magnitude less than 366. Going back to the pack of cards, it's basically the same problem with 52! days in the year and not 366. I tried to compute the 50% probability, but the numbers are still so big that I'm reaching the limits of software and hardware capabilities. I'm conjecturing that it's orders of magnitudes smaller than 52!, something like billions of times smaller. If anybody has a workaround (or a quantum computer) to calculate that I'd be very interested; there's gotta be a way. Something else to consider is that Steven Fry seems to assume here that every shuffle ever performed perfectly randomized the deck of cards being shuffled. That's unlikely to say the least. It's very probable that some combinations showed up many many times already. Maybe following a power law distribution, not sure about that. I'm not a mathematician, so if you see any flaw in my reasoning please let me know. Thanks for reading!

If you had a new deck of cards, the deck is in a standard order. Ace Hearts, King Hearts, etc. So the first time you shuffle that deck, the chances are greater that you would have a repeat shuffle.

So, written out in non-numeric form, the order of a pack of 52 standard playing cards could be any one out of eighty unvigintillion, six hundred and fifty eight vigintillion, one hundred and seventy five novemdecillion, one hundred and seventy octodecillion, nine hundred and forty three septemdecillion, eight hundred and seventy eight sexdecillion, five hundred and seventy one quindecillion, six hundred and sixty quattuordecillion, six hundred and thirty six tredecillion, eight hundred and fifty six duodecillion, four hundred and three undecillion, seven hundred and sixty six decillion, nine hundred and seventy five nonillion, two hundred and eighty nine octillion, five hundred and five septillion, four hundred and forty sextillion, eight hundred and eighty three quintillion, two hundred and seventy seven quadrillion, eight hundred and twenty four trillion possible combinations. And if you think that wall of text is big, just imagine the sheer enormity of ALL of those card order combinations if they were layed out next to each other. oO'

while this makes sense, there is no way of knowing that he didn't shuffle those cards the exact same way as someone else has..

Guys you don't appear smart because you yet again point out that it isn't 100% certainty but instead 99,99999% You don't have to mention it 52! times.

It's about the same odds as me winning the bloody lotto......and that's HIGH.

Quick wolfram alpha calculation shows that the average number of shuffles you would need to go through every possibility is way larger - I got 1.2101169951...e67

not mathematical certainty, it is probability.

My next question would be......... If you took all the card games in all the casinos from all over the world, found out how many times packs of cards were shuffled per day in total on average. How many days/years/centuries would it take at most to achieve all combinations?

That’s Quite Interesting.

it's highly improbable but not impossible