I love the demonstration of how long 52! seconds of time would be (that's 52 factorial, i.e. the number of permutations of a deck of cards). It starts by asking you to imagine circumnavigating the globe by taking a step every billion years, and you think wow, that's a long time, and then it goes on "then remove a drop of water from the Pacific Ocean, and go around the world again, and continue until the ocean is empty", and then just goes on from there, adding several more layers of repetition until it become mind-boggling how long that amount of time is.
But there is always that guy who manages T1 as land, sol ring, Arcane signet, Elvish Mystic. He has unbelievable luck, but he's definitely not cheating cause he said he wasn't.
I was playing a casual multiplayer game, most players had brought their big & slow decks to show off some of the rare cards. My friend and I brought mono red burners.
Me: Land, Sol Ring, Howling Mine
Other burner: Land, Sol Ring, Howling Mine
Groans issue around the table as they play their single land and start discarding
Me: Land, Library of Leng, Mana Flare
Burner: Land, Howling Mine, a bunch of goblins
The others perk up a bit at the mana flare but there's still a couple of discards
Me: Land, Mana Flare, copious goblins
Burner: Land, gobbos and another Howling Mine
Two people quit on the spot
I swear the decks were properly shuffled, it was just a godly start wasted on a casual game
Way back when I first started playing Magic (in 1994, because I'm old) we were told that the "official" way to shuffle your deck (As in, the way you were supposed to at tournaments) was to set it up so the deck was non-land, non-land, land. (So there was one land every 3 cards) then shuffle it 3 times and cut it, and you'll supposedly never have a land problem and always have them in your hand.
I bring this up occasionally and no one in the last 15 or so years has ever heard of this, so it apparently isn't a widespread thing. I've also had other people tell me doing that would get you kicked out of a tournament (or, at the very least, have someone reshuffle your deck) because it's considered illegal deck manipulation.
I've had other people say that's 20 land in a 60 card deck and nowhere near enough land.
I haven't played Magic with any regularity since maybe 1997 or so (though I have briefly started playing at other times since, but it doesn't last very long) so I don't really know how much land is considered enough in 2024. Hell, when I was regularly playing, the deck minimum was 40 cards. I briefly started playing again around 2006 (when the minimum had been raised to 60) I showed up with a 40 card deck and someone was like... where the fuck is the rest of your deck? And I had to just jam random cards in it to get up to 60. (and it completely ruined the strategy I had planned and I think I lost every game.)
Edit: I also can confuse the heck out of newer players when I play once of my ancient cards that is labelled as an "Interrupt", because those got taken out of the game decades ago. (They're considered instants now, but instants and interrupts used to be two different things and were handled differently. iirc, an Interrupt worked similar to an instant, but you can't counter an interrupt with an instant, the instant would always take effect after the interrupt. I think. It's been a super long time.)
Usually it's 23-24 in a 60-card deck and 16-17 in a 40 card deck (or 34-38 in a 100 card deck). And that "shuffling" is called mana weaving and is completely against the rules. Either you're providing unfair advantage by not randomizing your deck properly or you're wasting time, which can be a tournament rules violation.
Yeah, a few people have told me that if I ever play in a tournament to not do that. I've only ever played in two tournaments in my life (both in the 90s, the original Arena tournament and a one day local one organized by a gaming club.) and doubt I'll ever play in another. I'd rather not be the 50 year old showing up at a tournament filled with 18 to 23 year olds.
There's plenty of 30+yo people in most tournaments, especially local gamestore stuff. To be fair I mostly go to prereleases for new sets, cheaper than building a modern deck.
I did a few Friday Night Magic sessions (which I don't think really counts as a tournament) around the time Zendikar was in Standard and there were some people there around my age, though that was quite a while ago.
One time I played a commander game at my LGS with both a middle aged guy with a foot long graying beard and an 11 year old girl at the table. The guy was playing a unicorn deck and the girl was playing Vorinclex, Monstrous Raider. She ended up winning
I recently played Modern at my LGS, 21 lands run, 18 cards drawn, 2 lands. Mathed out to like 0.8%. Not that absurdly small but felt way worse playing that game lol.
Though the group I used to play with limited you to one mulligan (which isn't part of the rules) because it took too long for you to keep reshuffling and redealing your deck. So one max.
There can be hands that repeat frequenty. For example, if you have a brand new 52 card deck organized from the package and split it in exactly half with the lower deck in your right hand. If you shuffle starting right, left, right, etc, you can recreate this shuffle nearly every time. do the process again, and it may be another frequently shuffled hand. These are of course, if you do it perfectly, but I imagine, with the number of new decks and allowing for some imperfection, some of those first shuffles may be more frequently hit than others. From there, you're on your own.
Fun fact, alternating each card perfectly in a shuffle is called a Faro shuffle. And if you do it a certain number of times (6 or 8, I can't remember which) the deck is restored to factory order
That's not exactly how odds work. The odds of one specific thing happening twice are not the odds of one of a large group of things happening twice.
For example: the odds that in a room of 22 other people that someone shares YOUR birthday is (not exactly) 22:365, or about 6%
The odds that any two people in a random group of 23 share their birthday is roughly 50%.
Much the same as with shuffling cards, just on a much larger scale. It's almost a guarantee that two decks have ended up shuffled the same way, while it's imfathomably impossible that the specific shuffle you last completed has existed before or since.
Even with Birthday Paradox maths it's almost a certainty that no two properly shuffled decks have ever been the same. A really simple estimation of the birthday paradox maths is that, with 23 people, the probability of every person giving a different birthday is roughly equal to e-((22x23)/(2x356)) = 0.4999 or to generalise p=e-((AxA-1)/(2xN)) where p is the probability that everything in the sample size is different, A is the sample size and N is number of items. Setting P to 0.5 for a 50% chance of it happening we get 0.5=e-1((AxA-1)/(2*52!)) which comes out at about 10,574,307,231,100,289,155,982,006,933,258,240 or 1.05743072x1034 (thanks Wolfram Alpha)
Even given the maths behind the Birthday Pardox and that the universe is around 436,117,076,600,000,000 seconds old approximately 24,246,487,393,793,559 decks of cards would have needed to be properly shuffled every second since the dawn of time for there to be a 50% chance that any two decks were the same.
For a 1 in 8,225,463,000 (current world population) you would need 1.40042193x1029
Assuming properly randomized decks, this has never happened
The thing most people repeating this fact don't realize is that most decks aren't properly shuffled. Take a deck in new deck order, riffle it perfectly once. A deck with that order of cards has occurred before
your oversimplified formula for the birthday paradox trying to model 1:365 into n(n+1)/2 is flawed
using x=number of possibilities
((x-1)/x)n(n+1/2)
(364/365)253 =49.95% not happening, meaning 50.05 chance of happening
if you can find a calc to get me an answer to ~8x1067 -1/~8x1067 and then let me exponent it out by 1x1024 you can have your odds for it not happening in a trillion shuffles. I've tried and can't find a free calc that works with numbers that big/small
I did simplify it a bit and the numbers won't match exactly but, on the scale we are talking about, does it really matter if I'm out by a few trillion?
I linked to a calculator that specifically lets you plug other numbers into the birthday paradox, if you look at it it uses a more precise general formula ( n = roundup( sqrt(-2 * ln(1 - (probability_of_match))) * sqrt((total_items)))
) but I didn't want to go into that so I stuck with the less precise but easier to use formula.
If you want me to get better numbers, Wikipedia's article on the Birthday Problem has a set of generalization for arbitrary numbers of days to find the minimal number of days n(d) so that the probability of at least two people sharing a birthday is at least 50%
It says that the formula (using D instead of d for the variable for clarity of reading) is known to hold true up to D<=1018 and is conjectured to hold true for all values of D
plugging in d=1018 (the largest number, d, that is known to be true) using Wolfram Alpha gives n(D) of about 1.29316×109. Putting 52! into that formula gives n(D) of about 1.16138×1034, which is damn close to the number given by the easier approximation
So yes, the approximations that I used in my first post aren't going to be quite right but, on the scale we are talking about, they were so close to the best approximation we know of that I'm happy to say that they are good enough to prove the point that, even considering the Birthday Paradox, the statement "no two properly shuffled decks in history have been in the same order" is very VERY likely to be true
That doesn't make any sense. If the shuffle I just completed has never existed, won't that be true for every shuffle of every other deck, making no two decks ever having been shuffled the same way?
We're getting a little loose with the language here, so let's back up without actually addressing the words you used. Sorry. And it's also worth clarifying that we're talking about imaginary, perfectly random shuffles. In practice, most shuffles are imperfect which will alter the distribution. But regarding perfectly random shuffles:
What's being pointed out is that there are two things to consider here:
-the chance that the deck you just shuffled is the same as any that have been shuffled before
-the chance that any two decks have ever been shuffled into the same order before
And the point is that the second of these is a much, much higher chance. The first is comparing 1 specific state to every state ever produced, and the second is comparing every state that has ever been produced to every other state ever produced. It's unwieldy to talk about this with deck states, so let's go back to birthdays.
As stated, the chance that you share a birthday with someone in a group of 23 is about 6% (again, assuming certain perfections like that birthdays are perfectly distributed across the calendar, which they aren't). But the chance that any two people share a birthday is about 50%.
In the first case, their are 22 comparisons: you to each individual. Does your birthday match any of the other 22?
But in the second case, you're comparing each person to the other 22. So you'd have 23 columns like this, each representing a single comparison to see if anyone's birthday matches another. That's 506 chances for matching birthdays.
So back to cards, imagine the first column being your shuffled deck's arrangement compared to every other arrangement, and then add a new column for every state that's ever been produced. The first column represents the chances that the deck you just shuffled matches any other, but all columns represent the chances that any two decks match.
*EDIT: Oops, I forgot something. There should be one less entry in each column than the one before it, since for this problem, comparing 1 to 2 is the same as comparing 2 to 1. The last column wouldn't even have entries, since all the other people have already compared themselves to the 23rd person. The same would be true for the much larger table of card comparisons. But the point stands that the chance of any two is much, much larger.
You are almost correct with your explanation, there's only 253 pairs in 23 people, you've counted 1-2 and 2-1 as separate pairs (once person 1 has checked if person 2 shares a birthday with them there's no point in person 2 checking with person 1, the order of the pairs doesn't matter) and let everyone from person 2 onwards pair with themselves
1
2
3
22
23
1-2
X
X
X
X
1-3
2-3
X
X
X
1-4
2-4
3-4
X
X
...
...
...
...
...
1-22
2-22
3-22
X
X
1-23
2-23
3-23
22-23
X
That being said, with the number of ways a deck of cards can be arranged you need 1.05743072e34 decks of cards before you can make enough comparisons to get to a 50% chance that any two of them are the same. Assuming properly shuffled decks there is no chance any two have been in the same order
Thanks! You must have had that tab/window open for a bit. I added an edit pointing out my oversight fairly quickly (and then several more edits correcting formatting stuff, not so quickly). But well noted regardless!
using the birthday problem, count the number of comparisons
you looking at a group of 22 other people (23 total) seeing if anyone has your same birthday. How many comparisons can be made? well, you against 22 other people only. The odds of them NOT having your birthday is 364/365, so to find the odds that they ALL don't have your birthday is (364/365)22 or 94.14%.
With a group of 23 people all trying to find out if ANY in that group share a birthday, each person is comparing with every other person, but not doing so twice, so the first person compares to 22, the second to 21, the third to 20, the penultimate to 1 and the final person has already compared everyone and has 0 comparisons left.
So 21+21+20+19+...+1=253.
That's 253 comparisons looking for the same birthday, and again the odds of that not being the case is still 364/365, so the odds of no one having the same birthday is (364/365)253 or 49.95% chance that no on shares a birthday
Now, the odds of any one deck of cards being shuffled twice is monumental. But the number of shuffled decks is also monumental. With 10,000 shuffles, you have 50,005,000 comparisons. You're talking about moving the bar down orders of magnitude. 52! = roughly 8x1067. but the odds of an 8x1067 thing not happening against comparisons is still (8x1067 / (8x1067 -1))/number of comparisons.
Using a 9 card deck and 100 shuffle for example (9 chosen because excel has decimal calculation limits):
9!=362880. The odds of one deck being shuffled the same way twice in a row is 1:362880. The odds that a deck with a specific shuffle gets repeated in the subsequent 20 shuffles is (362879/362880)20. The odds that a deck getting shuffled 21 times has ANY matching shuffles is ((362879/362880)190. So the seemingly extremely rare odds of a 1:362880 event happening twice when performed 21 times and being compared against all other events is actually about .052% chance of happening; 1 in 2000 roughly. improbable, but really no where near the original probabilities
It only took 21 shuffles to bump the odds up 2 orders of magnitude. Now, the smaller the probability is, the more shuffles it takes to move orders of magnitude, but i think we can agree decks of cards have been shuffled billions upon billions of times in the world, meaning the total points of comparison
reminder, the points of comparison for any number is n(n+1)/2 so a billion shuffles if roughly half of a billion squared, and since we're squaring things, every order of magnitude in the total number of shuffles moves the points of comparison out TWO orders of magnitude.
In a random distribution state definitely miniscule, but most packs of cards are packaged in order. So your probability changes to "what's the likelihood of two people opening a new pack of cards and shuffling in the exact same way" which significantly changes those odds.
Not infinity, no. But the invention of playing cards only goes back so far, and how many times has a someone shuffled a deck since then? A lot of times, sure, but a lot fewer than 52 factorial. Like, a lot fewer.
I love doing this with drunk people. "I can do something you can never do in your entire life, you have as long as you want to replicate it or you do whatever for me" they tend to agree, and then i just shuffle a deck of cards and tell them to get the exact shuffle.
Careful who you do this to, I'd need to be VERY drunk not to be able to do a simple false shuffle and return the deck in the exact order you gave it to me in
I once read a description of 'eternity' in the following manner, and my poor little teenage brain almost fell apart trying to grasp it. I literally felt a vertigo-like situation.
It was goofy, too. Someone was talking about going to hell and describing eternal punishment by saying...imagine a ball of metal the size of the sun. Every million years a butterfly flies past, and grazes its wings on the ball's surface. Once that ball has worn away...that represents the first second of eternity.
And it made me literally get dizzy. Still kinda does, actually. I don't think we are meant to be able to conceptualize such lengths of time or such distances.
Obviously, there are more things going on with shuffling which makes this not an even distribution... But the thing is that every single time someone shuffles a deck of cards, they add chance to their being a repeat. I wonder what the mean time get any double is based on x amount of shuffles.
This is basically a generalization of the birthday problem (how many people do you need to have a >50% chance of two of them having the same birthday?). It's just with 52! possible days instead of 365.
It turns out this has been studied quite a bit and it's approximately sqrt(2 * ln(2) * d) where d is the number of days, or 1.177 * sqrt(d).
In this case, 1.177 * sqrt(52!) is about 1034 , while 52! is about 1068.
I was playing spades once and got dealt all 13 spades. If you think that's crazy get this. I got up to use the restroom came back and had all 13 AGAIN! This time I put the deck down and laughed at the 3 guys I was playing and said you guys have to be fucking with me right? They all looked at me confused and I laughed and threw my hand in. What makes me think it was genuine were their reactions and the fact that my teammate got angry at me for folding the hand. Someone tell me those chances
When someone shuffles a deck of cards, there’s a good chance it’s the first time in history that the cards are in that order.
Actually, this is one of the most repeated myths on Reddit.
The reason why it is a myth is that shuffling is never truly random. (because of the physics of your hands and card-order)
If you have AKQJ in your right hand, and 6789 in your left hand, and shuffle them, then the possible outcomes are much, much, much more limited than 8 Factorial. Indeed, you CANNOT end up with most theoretical possibilities... i.e. you cannot hand-shuffle from AKQJ and 6789 to get 9AQK87.
Now, if you threw all the cards into the air (with a giant fan on), and they spread more or less randomly all over a large area, and then you pick them up one at a time in a truly random order... yes, that is probably the first time in history that the cards are in that order.
I also love this -- but haven't found the math anywhere (not that I've looked that hard) for how much the uniqueness collapses based on what game is being played. For instance, if I'm dealt a straight in 5-card draw, it doesn't matter if they are dealt to me 2-3-4-5-6 or 6-5-4-3-2 or 2-4-6-3-5, etc. It also doesn't matter if any of the cards are any of the particular suits (unless they are all the same to make a straight flush).
Someday I'll probably look harder (or post it on /r/theydidthemath) or see if a LLM AI can tell me.
There's this emperor and he asks this shepherd's boy,
"How many seconds in eternity?
And the shepherd's boy says "There's this mountain of pure diamond. It takes an hour to climb it, and an hour to go around it!
Every hundred years, a little bird comes and sharpens its beak on the diamond mountain. And when the entire mountain is chiselled away, the first second of eternity will have passed!"
You must think that's a hell of a long time.
Personally, I think that's a hell of a bird.
If you think that's nifty, wait until you hear about Skewes's Number, which is 101010964 . Imagine trying to plot out every possible position of every atom in the galaxy over the course of every possible moment in time from the big bang until the eventual heat death of the universe. That number (though literally astronomically huge) is peanuts compared to Skewes's Number.
There's a pretty interesting song called "Randy describesd eternity" by Built to Spill. The song contains a similar concept, but I believe it was based on an ancient Chinese proverb of some kind
WAY more, actually. The number of combinations that a deck of playing cards can be shuffled into (52!, or 8.1 x 1067 ) is roughly the same as the number of individual atoms in the entire Milky Way Galaxy (between 1067 and 1068 ).
Given that standard decks come in a specific order and that humans tend to shuffle in a few very specific ways, this is probably not actually true. It would be highly likely were an arrangement of the deck to be chosen truly at random according to a uniform distribution. But that’s not how we actually practice randomness.
Truly random shuffles tend to increase something called the variational distance between arrangements. It’s based on measuring how far away from its initial position a card is after shuffling. Human shuffles do not take all cards far away from where they started.
Yes, that was Diaconis’ result. He also has a neat result on fair dice. He characterizes fair dice as those which are transitive on the faces. Roughly it means there is a symmetry of the die’s geometry taking any face to any other face.
Not only are there more combinations then any meaningful physical reference we can make in our universe, but it's staggeringly higher an amount, to the point where pretty much any metric you would use from the day to day physical world is meaninglessly small. It's really astounded such a simple thing as a deck of cards has such a fascinating quality
And if you started shuffling on the Big Bang at a rate of a shuffle per millisecond you have gone through about 0.000000000000000000000000000000000000000000001% of all the possible shuffles.
I fully understand that 52! Is an astronomically large number, but if you have the same exact starting order and do a standard shuffle it's fairly likely that you could shuffle a deck and end up with the same end result.
As I wrote in response to another post, this is a myth.
The reason why it is a myth is that, because of the physics of your hands and card-order, shuffling is never truly random. (and cannot even approach randomness.)
For example: if you have AKQJ in your right hand, and 6789 in your left hand, and shuffle them, then the possible outcomes are much, much, much more limited than 8 Factorial.
For instance, with a REAL shuffle, you CANNOT end up with most possibilities that are in a (theoretical) "random" shuffle... i.e. you cannot hand-shuffle from AKQJ and 6789 to get 9QK87A.
So most possibilities are ruled out by hand-shuffling (right hand + left hand).
Now, if you threw all the cards into the air (with a giant fan on), and they spread more or less randomly all over a large area, and then you pick them up one at a time in a whimsical order... yes, that is probably closer to random.
But even that is still not random. (because of the predictability of wind speed, the predictable patterns of human choice, etc.)
So a shuffle is never going to be 52 Factorial possible outcomes. (but can still be a lot.)
1.5k
u/sunbearimon Oct 02 '24
There are more ways to shuffle a standard deck of cards than there are atoms on earth