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[jonts26]

Also, if jonts26 wants to bust some mathy statsy goodness, I would definitely appreciate it.

Well, I don't have much to add the the previous few posts. A lot of the problems here seem to stem from how we define random. Which I guess is a hard thing to pin down and I don't feel like attempting now. Anyway, here are some random thoughts and fun facts!

The purpose of dealing 1 at a time is not to further randomize the deck, but because it's harder to cheat that way since most deck stacking techniques rely on forced clumping of cards during the shuffle. It is much, much easier to stack the deck in euchre than bridge.

There are indeed 1/52! card combinations. That is a lot. For perspective, if you start with a random deck, every time you give it a shuffle, you have almost certainly created a deck ordering which has never happened before. That still blows my mind.

@theorel: The wannabe philosopher in me wants to point out that maybe there is no such thing as random and everything is deterministic.

After 8 perfect shuffles, you return a deck to its original ordering.

[DStu]

First, thanks qmech, I think most of I wanted to write is not neccessary any more (or yet, lets see how the thread goes).

A "uniform probability distribution" means that given a point in the deck each card has a 1/52 probability of appearing there.  Another example of a uniform probability distribution is a non-weighted die, which has a uniform probability distribution for all its numbers (1/6).  This is a fully randomized deck.

It's even stricter. You don't only want to each card to have it's propor position, you want to have each of the 52!=8x10^67 possible permutations of the deck is equally likely. And 10^67 is a large number, the sun has 10^55 atoms (accorting to Wolfram Alpha). So that's nearly a trillion times the number of atoms in the sun.
Do you really come near this distribution? I don't know, it's hard to tell, it's really hard to generate significantly more than 10^67 shuffles to compare the outcome with the uniform distribution. At least if you are limited by the lifetime of the universe...

The good thing is that, when playing most card games, you are not really interested in the exact permutation. You care for what cards come to your hand. So when you distribute your 52card deck to 4 persons, each one getting 13 cards, you don't care in which cards they get first, what matter is which one come to their hand. That mean out the 8x10^67 permutations, you map (13!x13!x13!x13!)=1.3x10^39 permutations to one and the same hands.

And as described above, you can either use this large contraction to try to cancle out any distance from the uniform distribution of all hands, or you you can leave it. Typical shuffles tend to leave cards that where close before the shuffle also close together after the shuffle. Which means, the probability of permutations where these two cards have a smaller distance from each other is 'a bit' larger than it should be, and the ones where they have a large distance is 'a bit' smaller. Where 'a bit' depends on how often you shuffle, but is always larger than 0.

So you want to give each player cards from all parts of the deck to compensate for this behaviour. If you don't, and A and B where next to each other before the shuffle, and he has card A, it's 'highly' likely that he also has B. At least more likely than it should be. If you do, it's significantly lower. See example (with extreme numbers for demonstration effect) above.

[Davio]

Actual randomness: Every permutation is possible with the same likeliness. The chances of the deck being A-2(diamond),A-2(heart),A-2(spade),A-2(club) is the same as 2-A(diamond),2-A(heart),etc and the same as specifically Ac3s8d4h5s....

Human randomness: No two cards of the same suit and a rank higher or lower should be near each other.

In random distributions, clustering (grouping) will appear.

[Kirian]

* Unless you're playing euchre, where you deal 2/3/2/3 cards and then 3/2/3/2 cards at a time.

I play quite a bit of euchre with my family, who learned in Ontario and we've never heard of 2/3/2/3 and just deal 1 to each person till everyone has 5 leaving 4 cards which you check before flipping to make sure you didnt misdeal

Crazy Canadians.  Here in OH it's 2/3, and you'd damned well better count the four cards left before flipping.  Also, no "partner's best" BS, if you're playing alone, play alone.

[Davio]

In the Netherlands with the game "Klaverjassen" the de facto standard is 3-2-3.

In fact, when playing with an older crowd in a competitive setting (tournament), the players can get pretty mad if you divert from this standard. Those players can also get mad when someone grabs their cards before everyone has 8. They can be pretty tough, those Benidorm bastards.
And if you shuffle too much you even get the complaint that "you're shuffling the faces off the cards". 

[Kuildeous]

I play quite a bit of euchre with my family, who learned in Ontario and we've never heard of 2/3/2/3 and just deal 1 to each person till everyone has 5 leaving 4 cards which you check before flipping to make sure you didnt misdeal

I never played euchre. I only learned about it about 3 years ago when I did a show that takes place in the UP ("that's Upper Peninsula for all ya flat-landers"). I researched the game on Wikipedia.

I get the impression that the traditional game sticks with 2/3/2/3, but I'm sure more casual players don't worry about it. Kind of like how pre-Hinterlands, our group would draw the first hand, know what the second hand was and just combine our first two purchases. It would have bugged a Dominion purist ("What? But why would you buy your second card without knowing what the other players are planning?!"), but it worked for our casual group.

[WanderingWinder]

Actual randomness: Every permutation is possible with the same likeliness. The chances of the deck being A-2(diamond),A-2(heart),A-2(spade),A-2(club) is the same as 2-A(diamond),2-A(heart),etc and the same as specifically Ac3s8d4h5s....

Human randomness: No two cards of the same suit and a rank higher or lower should be near each other.

Yeah, but this is the problem. That should happen sometimes.

[WanderingWinder]

Right but 'random' does not mean 'the uniform distribution'. In fact, a uniform distribution is highly ordered, very much not random. Which is the crux, I think, of our misunderstanding.
Random basically means without a pattern. There either is a pattern, or there isn't. I don't see how you can 'almost' or 'nearly' have a pattern. If you want to enlighten me, please go ahead.

Good news: the problem here is only one of terminology.  The uniform distribution on the set of all possible deck orderings is the one that assigns each one an equal probability.  There seem to be a lot of mathematicians around here, who don't think twice about using this (very useful) terminology, but thinking it refers in some sense to spreading copies of the same card out evenly is totally understandable.

Indeed, I do math work at a university, I'm quite aware of what the formal definition of the uniform distribution means. I just thought that you couldn't possibly be referring to the uniform distribution on the set of all deck orderings, because this is impossible to get.
At any given moment in time, the deck has one, particular, distinct order. The probably distribution is 1 that it is in this distinct order, and 0 that it's in any other order. I guess you can say that the deck has achieved this uniform distribution if you put it into a process that leaves you with equal probabilities of every ordering (even this is a stretch really), however, Im pretty sure that such processes don't exist. Though, if you believe the Copenhagen quantum mechanics people (and I don't, but I'll humour you), such a process could exist, but I'm fairly sure the amount of time it would take would take is going to be well longer than the amount of time you'd expect the cards to exist without breaking down, being destroyed, or at least becoming indistinguishable from each other. So, yeah....
I stand by my assertion that for all intents and purposes, the randomization of card decks refers to the inability of humans to predict which card comes at any point, better than giving random equal weight to every card they know to remain in the deck. For most card games, anyway.

[DStu]

But you can model many processes to be random, and even if it is deterministic on some layer, the model will represent what you know about the process. If you don't know how the fingers moved, you can as well assume the distribution is "uniformly random", and probably that is your best guess on how to describe you knowledge of the state of the deck.

It's deterministic in some state I don't know, and all states are from my perspective equally likely is perhaps true, but what do you get from the fact that it is (maybe) deterministic. You know as much about it as if it where random, so you can model it as being random.

It's kind of like, for every-day life, not using the 'fact' that our enviroment can be modelled to be R^3, because Heisenberg uncertainity, and Einstein space curvature, and thus, so it's not "true" that the space is R^3. But you will get to work much easier if you assume it is.

[WanderingWinder]

But you can model many processes to be random, and even if it is deterministic on some layer, the model will represent what you know about the process. If you don't know how the fingers moved, you can as well assume the distribution is "uniformly random", and probably that is your best guess on how to describe you knowledge of the state of the deck.

It's deterministic in some state I don't know, and all states are from my perspective equally likely is perhaps true, but what do you get from the fact that it is (maybe) deterministic. You know as much about it as if it where random, so you can model it as being random.

It's kind of like, for every-day life, not using the 'fact' that our enviroment can be modelled to be R^3, because Heisenberg uncertainity, and Einstein space curvature, and thus, so it's not "true" that the space is R^3. But you will get to work much easier if you assume it is.

This is my point almost exactly (though you can model anything to be anything... ) The point is that people don't know it any different from random. Which is not the same as random, if you want to be precise. But it cuts against the 'almost' random thing. Because either you have some idea as to what cards are where, or you don't. Binary.

[DStu]

This is my point almost exactly (though you can model anything to be anything... ) The point is that people don't know it any different from random. Which is not the same as random, if you want to be precise. But it cuts against the 'almost' random thing. Because either you have some idea as to what cards are where, or you don't. Binary.

It's not 'almost random' is 'almost uniformly random', which the 'almost' bounded to 'uniform', not to random. So it describes a random distribution, which differes a bit from the uniform distribution. (Or some deterministic state which people don't know different from a distribution which is almost uniform).

(But I like to appreviate 'deterministic state the people don't differen from random' by 'random state' in the future, if you don't mind, also I see that there might be some interesting philosopical question (which I don't find interesting) behind it)

Edit: I also see that people have said "nearly random", but this should be parsed to "nearly uniformly random", or maybe "nearly in the obvious distribution" (in case the obvious one is not the uniform distribution).

[WanderingWinder]

This is my point almost exactly (though you can model anything to be anything... ) The point is that people don't know it any different from random. Which is not the same as random, if you want to be precise. But it cuts against the 'almost' random thing. Because either you have some idea as to what cards are where, or you don't. Binary.

It's not 'almost random' is 'almost uniformly random', which the 'almost' bounded to 'uniform', not to random. So it describes a random distribution, which differes a bit from the uniform distribution. (Or some deterministic state which people don't know different from a distribution which is almost uniform).

(But I like to appreviate 'deterministic state the people don't differen from random' by 'random state' in the future, if you don't mind, also I see that there might be some interesting philosopical question (which I don't find interesting) behind it)

But almost uniformly random doesn't mean anything if uniformly random doesn't mean anything, and uniformly random doesn't mean anything if you don't take the strict mathematical definition, and if you DO, then you aren't 'almost' there at all - you either are or aren't. Again. But we're arguing in circles here.
I do mind you calling that random, but 'seemingly random' or 'apparently random' seem fine to me.

[DStu]

But almost uniformly random doesn't mean anything if uniformly random doesn't mean anything, and uniformly random doesn't mean anything if you don't take the strict mathematical definition, and if you DO, then you aren't 'almost' there at all - you either are or aren't

First, uniformly random means quite a lot, if there is a question than if the deck is in this state.

I'm in this field for some years now, and my impression is that the mathemtics quite elgantly ignore this philosphical question of trying to adress "what is 'random'", so I would be curious to see this definition, where 'uniform random' does not mean anything.

Concerning the 'almost', it depends on the definition of almost. I remeber this also already to be in the thread, but there is no problem to define a metric for probability measures, and than "almost" means "some probability measure with small distance with respect to the metric".
If you like to have the discrete metric (you are either there or not, have either distance 1 if you are there, and distance 0 if you are not), than I wonder why you want to use it for probability measures, but not if you talk about your distance on your way home. You can't be almost at home, you either are there or are not.

[DStu]

Doublepost, want to understand where the terminology differ: maybe here?

Because either you have some idea as to what cards are where, or you don't. Binary.

Also if you have "some" information on the cards, I would still call them "random". A coin that shows head 99% of the time is still a random coin, just on which is quite "far" away from the ideal one that shows head 50% of the time. But I can not know the output until it falls, so it's random.

Pedanticly, also deterministic states are probability measure which just happen to have all mass on one state, but I accept to not use the word "random" when something is deterministic to avoid confusion.

[WanderingWinder]

But almost uniformly random doesn't mean anything if uniformly random doesn't mean anything, and uniformly random doesn't mean anything if you don't take the strict mathematical definition, and if you DO, then you aren't 'almost' there at all - you either are or aren't

First, uniformly random means quite a lot, if there is a question than if the deck is in this state.

I think this sentence got away from you. Certainly the second half is a conglomeration of words that doesn't mean much anything to me.

I'm in this field for some years now, and my impression is that the mathemtics quite elgantly ignore this philosphical question of trying to adress "what is 'random'", so I would be curious to see this definition, where 'uniform random' does not mean anything.

Did you fully read my post? I said it doesn't mean anything IF it doesn't mean the strict mathematical definition. I'm saying that is the only definition. I'm not saying that that is not a definition. But you can't ignore the philosophical question, and you don't grasp it if you think you can. The only way to try to 'ignore' it is to accept one side of the argument, which is of course not ignoring it at all. The whole philosophical question is what you mean when you say that. If you truly ignore it, you can't communicate.

Concerning the 'almost', it depends on the definition of almost. I remeber this also already to be in the thread, but there is no problem to define a metric for probability measures, and than "almost" means "some probability measure with small distance with respect to the metric".
If you like to have the discrete metric (you are either there or not, have either distance 1 if you are there, and distance 0 if you are not), than I wonder why you want to use it for probability measures, but not if you talk about your distance on your way home. You can't be almost at home, you either are there or are not.

I reject the notion of the distance metric for a concept such as randomness. Similarly, something is either chaotic or not. Jimmy is either in love with Alice, or he isn't. For certain things, distance makes sense. For others, it does not. I am positing that randomness belongs to the latter class, rather than to the former.

[WanderingWinder]

Doublepost, want to understand where the terminology differ: maybe here?

Because either you have some idea as to what cards are where, or you don't. Binary.

Also if you have "some" information on the cards, I would still call them "random". A coin that shows head 99% of the time is still a random coin, just on which is quite "far" away from the ideal one that shows head 50% of the time. But I can not know the output until it falls, so it's random.

Pedanticly, also deterministic states are probability measure which just happen to have all mass on one state, but I accept to not use the word "random" when something is deterministic to avoid confusion.

The coin that shows heads .99 is of course random around its .99 mean. But the deck of cards is not random about any mean state. It's a false analogy. The more apropos analogy is to a coin which is normally .5 like a normal coin, but given the circumstances of a particular toss, you're able to determine say a .7 chance it lands heads. For this particular toss, the results will be random about a .7 mean of course, but this is certainly not a 'random' toss of the coin.

[DStu]

Concerning the 'almost', it depends on the definition of almost. I remeber this also already to be in the thread, but there is no problem to define a metric for probability measures, and than "almost" means "some probability measure with small distance with respect to the metric".
If you like to have the discrete metric (you are either there or not, have either distance 1 if you are there, and distance 0 if you are not), than I wonder why you want to use it for probability measures, but not if you talk about your distance on your way home. You can't be almost at home, you either are there or are not.

I reject the notion of the distance metric for a concept such as randomness. Similarly, something is either chaotic or not. Jimmy is either in love with Alice, or he isn't. For certain things, distance makes sense. For others, it does not. I am positing that randomness belongs to the latter class, rather than to the former.

So things are either random or they aren't. So far we agree, I think. Now if they are random, (or if I model them as random because all I know of them is as much as I know of some random distribution), there are different probability distributions. There is the uniform distribution, and there are other distribtions. These distributions can have a distance.

So now I can model shuffling. This model is how I imagine shuffling to work, so in a sense that is "what I know" on shuffling. Under this model, I can calculate the distribution of the permutations of the cards. Again that is all I know on the state of the cards, so I can as well asume the cards are in the random state with the distribution I just calculated.  This distribition now again has some distance to the uniform distribtion. So now I'm that bold that if this distance is "small", I would call my cards "almost uniformly random".

[DStu]

The coin that shows heads .99 is of course random around its .99 mean. But the deck of cards is not random about any mean state. It's a false analogy. The more apropos analogy is to a coin which is normally .5 like a normal coin, but given the circumstances of a particular toss, you're able to determine say a .7 chance it lands heads. For this particular toss, the results will be random about a .7 mean of course, but this is certainly not a 'random' toss of the coin.

Why not? With this knowledge, it behaves like a coin with a .7 mean? A coin with .7 mean you consider as random. Why shouldn't you use the same notion for something that behaves exactly the same?

A conditional probability is still a probability.

[WanderingWinder]

The coin that shows heads .99 is of course random around its .99 mean. But the deck of cards is not random about any mean state. It's a false analogy. The more apropos analogy is to a coin which is normally .5 like a normal coin, but given the circumstances of a particular toss, you're able to determine say a .7 chance it lands heads. For this particular toss, the results will be random about a .7 mean of course, but this is certainly not a 'random' toss of the coin.

Why not? With this knowledge, it behaves like a coin with a .7 mean? A coin with .7 mean you consider as random. Why shouldn't you use the same notion for something that behaves exactly the same?

A conditional probability is still a probability.

If I flip the penny I have sitting  next to my keyboard, I am flipping a particular coin rather than a random one. Same reason.

[DStu]

The coin that shows heads .99 is of course random around its .99 mean. But the deck of cards is not random about any mean state. It's a false analogy. The more apropos analogy is to a coin which is normally .5 like a normal coin, but given the circumstances of a particular toss, you're able to determine say a .7 chance it lands heads. For this particular toss, the results will be random about a .7 mean of course, but this is certainly not a 'random' toss of the coin.

Why not? With this knowledge, it behaves like a coin with a .7 mean? A coin with .7 mean you consider as random. Why shouldn't you use the same notion for something that behaves exactly the same?

A conditional probability is still a probability.

If I flip the penny I have sitting  next to my keyboard, I am flipping a particular coin rather than a random one.

And where's the problem in flipping a particular coin?

[WanderingWinder]

The coin that shows heads .99 is of course random around its .99 mean. But the deck of cards is not random about any mean state. It's a false analogy. The more apropos analogy is to a coin which is normally .5 like a normal coin, but given the circumstances of a particular toss, you're able to determine say a .7 chance it lands heads. For this particular toss, the results will be random about a .7 mean of course, but this is certainly not a 'random' toss of the coin.

Why not? With this knowledge, it behaves like a coin with a .7 mean? A coin with .7 mean you consider as random. Why shouldn't you use the same notion for something that behaves exactly the same?

A conditional probability is still a probability.

If I flip the penny I have sitting  next to my keyboard, I am flipping a particular coin rather than a random one.

And where's the problem in flipping a particular coin?

There isn't any problem with it. It's just not random.

[WanderingWinder]

Concerning the 'almost', it depends on the definition of almost. I remeber this also already to be in the thread, but there is no problem to define a metric for probability measures, and than "almost" means "some probability measure with small distance with respect to the metric".
If you like to have the discrete metric (you are either there or not, have either distance 1 if you are there, and distance 0 if you are not), than I wonder why you want to use it for probability measures, but not if you talk about your distance on your way home. You can't be almost at home, you either are there or are not.

I reject the notion of the distance metric for a concept such as randomness. Similarly, something is either chaotic or not. Jimmy is either in love with Alice, or he isn't. For certain things, distance makes sense. For others, it does not. I am positing that randomness belongs to the latter class, rather than to the former.

So things are either random or they aren't. So far we agree, I think. Now if they are random, (or if I model them as random because all I know of them is as much as I know of some random distribution), there are different probability distributions. There is the uniform distribution, and there are other distribtions. These distributions can have a distance.

So now I can model shuffling. This model is how I imagine shuffling to work, so in a sense that is "what I know" on shuffling. Under this model, I can calculate the distribution of the permutations of the cards. Again that is all I know on the state of the cards, so I can as well asume the cards are in the random state with the distribution I just calculated.  This distribition now again has some distance to the uniform distribtion. So now I'm that bold that if this distance is "small", I would call my cards "almost uniformly random".

While this is still not how I would look at things, it's eminently reasonable, and I don't think we really have substantive disagreements at this point.

[carstimon]

I think that you guys are thinking about two different things- the deck itself, and the process of taking a deck and shuffling.  When WW says

There's no such thing as a 'near random' deck. It's either random, or it isn't. Again, similar cards being spread out is NOT the same thing as random.

he's talking about the deck itself.  Whereas DStu is talking about shuffling.

WW, tell me if this is on track to what you're trying to say:  Imagine I hand you two decks of cards, and I ask you "Which is more random?".  This is not meaningful.

And I think DStu is saying this:  I shuffle one deck once, and I shuffle the other deck 17 times.  The second deck is "more random" in the following sense:  the probabilities of which-card-is-where are closer to that of the uniform distribution.

[WanderingWinder]

I think that you guys are thinking about two different things- the deck itself, and the process of taking a deck and shuffling.  When WW says

There's no such thing as a 'near random' deck. It's either random, or it isn't. Again, similar cards being spread out is NOT the same thing as random.

he's talking about the deck itself.  Whereas DStu is talking about shuffling.

WW, tell me if this is on track to what you're trying to say:  Imagine I hand you two decks of cards, and I ask you "Which is more random?".  This is not meaningful.

And I think DStu is saying this:  I shuffle one deck once, and I shuffle the other deck 17 times.  The second deck is "more random" in the following sense:  the probabilities of which-card-is-where are closer to that of the uniform distribution.

I think DStu and I understand each other pretty well. Less sure you do. Most of all, don't think it matters too much at this point.

[timchen]

Because either you have some idea as to what cards are where, or you don't. Binary.

This I don't agree. Can you not have some idea about where are some of the cards? Say you have a deck with stashes, as a trivial example?

What I see you are saying is that you either know completely about a deck (not random), or you know it from partially to none (random). That is sure binary, but a rather useless statement.

Going to the original discussion, different shuffle and dealing definitely matters. Different shuffle gives you different random distributions (which hopefully all converges to the uniform distribution if you shuffle infinitely many times), but with a finite number of shuffles probably some methods converge faster than the others.

Also depending on the game, we might not distinguish some ordering from others. Say in bridge, cards from 2 to 8 are not going to matter that much usually, but the suit always matters. Therefore, a shuffle/dealing that does not exchange positions of numbers in the same suit is more forgivable than a shuffle/dealing that does not exchange suits. In this particular game, since the cards in the same suit are more likely to be played together, it is beneficial to deal it one card at a time to compensate for insufficient shuffle.