Is Random really Random?

[Smmenen]

I wrote this up in January of 2005. 

In retrospect, as I read it over for the first time in months, what I say seems pretty clearly wrong.  I'm not quite sure how I could have been so convinced of a position at the time, but I now see the critical flaw that I didn't see before.

Nonetheless, I promised I'd post it and maybe it will stimulate some interesting discussion.  It isn't well edited, so you may have to work a little bit to understand what I'm saying. 

Magic Theory: Is Random really Random? 

What is meant by Random? 

Random does not mean that anything is possible.  In magic, knowing where a card in your deck is cheating.  Yet, if I have 4 cards in my library, three of which I know are Dark Rituals, and one of which is a Black Lotus simply because I have seen every other card, there is a 75% chance that my top card is a Dark Ritual, therefore I can say with very high certainty that it is a Dark Ritual.  If I said to my opponent “I know what my top card is,â€? is that cheating?  Even randomized, the probability of my top card being Black Lotus is low compared to the probability that it is a Dark Ritual.  Therefore, random is not truly “random.â€?  The chance that my top card is a Dark Ritual is not equal to the chance that my top card is Black Lotus. 

But, assuming this, what does random really mean?  Does it mean that if the chance of the top card of my library being Black Lotus is equal to the chance that the middle card is?

Some terms: Even Distribution, Normal Distribution, Uniform Distribution.

An Even Distribution would be a distribution of each of the multiples of the cards in the deck evenly distributed.  There is no variation from this distribution; therefore, there is no function.  X=0.

A Uniform distribution is the complete opposite.  There is absolutely no pattern whatsoever to the distribution of cards. 

A normal distribution assumes that there is a distribution of the Dark Rituals around an “even distributionâ€? with a variance and standard deviation which is well known in statistics.  Under a normal distrubtion one standard deviation encompasses 68% of all cases.  Two standard deviations encompass 95% of all cases. 

The Basics

The chance of drawing at least one Force of Will in your opening hand is 40%.  The chance of there being at least one Force of Will in the subsequent seven cards is also 40%.  So if you drew 14 cards instead of 7, wouldn’t the probability of drawing at least one Force of Will be 80%?   

Answer: No.  The short answer why not is that 21 cards would be 120% - which is impossible and absurd.  The long answer is that we are overcounting.  To see this, let me explain why:

The probability of drawing *exactly* one Force of Will in the opening hand of 7 cards is 36%. 
This looks like this:
P(A) = .36
Where A = the chance of drawing a Force of Will in the opening hand of 7.

P(B) = .36
Where B is the chance of a Force of Will being in cards 8-14.

What brings the 36% up to the 40% of having at least one FoW in the opening hand is the chance that you will draw more than one FOW.  The reason that 80% is too much though is because we are overcounting the case in which BOTH sets of 7 have a FOW.  Therefore, you have to subtract.

The rule for adding probabilities is:
P(A)+P(B) MINUS the intersection of the event of A and the event of B.  The 40% value includes the intersection of the event of A and B.  What I mean by that is that the P(A) takes into account the probability of the first seven cards having a FOW when the second do not as well as the probability of having a FOW in the first seven cards when the second seven do as well.  The P(B) does the same but inversed. 

The interesting thing about calculating the probabilities where you are asking for the chance of getting *exactly* one card instead of *at least* is that the probability peaks with 13 cards at 44% but then begins to dip because the chance of drawing multiples begins to outweigh the chance of seeing the first one in the additional draws.  For this reason, you don’t have the problem of adding probabilities that could go over 100%. 

In order to calculate the probability of drawing at least one Force of Will in your first 14 cards, you find that by adding four values:

1) P(A) (which is the event of drawing exactly one FOW in the first 7 cards) = .36
2) P(B) (which is the event of drawing exactly one FOW in cards 8-14) = .36
3) The chance of drawing more than one FOW in either the first seven or second seven cards.
4) the chance of there being a FOW in both the first seven and the second set of seven. 

Since (3) and (4) are difficult to figure out, we can do a little trick to figure it out.  The easier way to figure it out is to figure out the failure rate. 

The probability of NOT drawing a FOW in any hand of 7 is 60%.  60% * 60% = 36%.  The chance of NOT seeing a FOW in the top 14 cards is 36%.  In other words, the chance of seeing at least one FOW in the top 14 cards is 74%, which for reasons already explained, is less than the 80% which one would obtain by adding the probabilities of seeing at least one FOW from the first and second set of 7 cards.

The Dilemma

One of the questions I have is whether knowledge of any part of this will change the outcome.  It seems intuitive that if the chance of seeing at least one FOW in the top 14 is 74%, then if I know that my top 7 doesn’t have a FOW, then the chance of their being a FOW in the subsequent 7, or the other half of the population, isn’t 74%.  However, it also seems intuitive that it should be more than 40%. 

With four copies of a card in a deck, there should be one every fifteen cards, on average.  60/4 = 15.  What does that mean if the deck is uniformly distributed?  It would be a meaningless statement. 

This came up from a game scenario that was like this:

Lotus Petal,
Land Grant,
Dark Ritual,
Brainstorm.
3 irrellevant cards

The question is:
Should you Brainstorm before or after you have cast Land Grant, assuming that all the Land Grant does is shuffle.  The question is: if we shuffle, will that increase our odds of seeing another Dark Ritual with the Brainstorm?

The real example was this:

Black Lotus,
Lion’s Eye Diamond
Dark Ritual
Land Grant
Darkwater Egg,
Chromatic Sphere,
Brainstorm

With as good as this hand is, I figured that the top of the library is probably cantrips and bad cards.  To see a good mix of cards I’d prefer to shuffle and get the bad cards shuffled into the library.  In other words, I’d say that shuffling the library increases the chances of seeing another Dark Ritual with the Brainstorm.

Is this rational?   That’s what this article seeks to explain.

Cheating

The fact of the matter is that if you know where a card is in your deck you are cheating.  However, for reasons I’ve explained in the opening segment, you can have a very good idea about what the card your about to draw may be even if you have randomized your deck and haven’t peeked. 

The answer to my question depends upon whether the deck is distributed in a uniform manner or in something similar or akin to a normal distribution.  I believe that randomized decks are a flatter normal curve.  In mathematical terms, I believe that the function or the waveform probability distribution of Dark Ritual is not flat. 

Quantum Theory
The centerpiece of Quantum Physics as I understand it is the uncertainty principle.  The principle is that we can’t truly know where something is, and if we check, we change the results.  There are two philosophical interpretations of this principle that are diametrically opposed and they are as old as quantum theory.  Albert Einstein for the absolute 'this is how it is despite us not being able to observe the result.’  So instead of saying, because we can’t truly know where something is doesn’t mean that it isn’t there.  On the other hand,  Niels bohr argued for the indeterminate state – the idea that nothing really exists in a precise manner.  That is, quantum mechanics at its most basic is either not fully defined (the absolutist theory) or indefinable (the indeterminate theory).  Under the indeterminate theory, the table doesn’t exist except as a variation – a variance of the location of the table. 

There is a hypothetical called schrodinger's cat dilemma.  The idea is that a cat is gassed or poisoned and put in a box.  Is the cat dead or not? Einstein would say, the cat is either dead or it isn’t, but it is one or the other.  Bohr might say, the cat could be dead and alive, the important thing isn’t whether the cat is in fact dead or alive but the probability distribution.  Quantum theoreticians tend towards the latter Bohr position.   

How does this apply to magic?  We can’t know the precise location of any card in our deck, but if we agree with Bohr, our card exists as a probability distribution.  Our deck IS Shodinger’s cat. 

I want to be clear.  I am NOT assuming an EVEN distribution.  But I AM assuming that the deck will tend toward SOME distribution.  I think this is provable. 

I think that if you did a million goldfishes after a thousand shuffles for each goldfish, and divided the deck into four piles of 15, you would find a dark ritual in two piles more than 50% of the time.  I think it is likely that a very sizable portion of the time, probably over 40%, you would find a dark ritual in three piles.  Regardless, if we assume that there is one dark ritual in two piles, that is reason enough to shuffle, in my opinion.  Put more risk on it.  What if you had the choice to Spoils of the Vault for Dark Ritual when you already have one in hand.  Your options are to play it before Land Granting (assuming the risk that the land is removed is irrelevant and the thinning is irrelevant), or you can play it after Land Grant.  The point is: you have a Dark Ritual in your opening hand, which decision maximizes your life? 

I would argue that the simple assumption that there will be a Dark Ritual in two piles over half the time assumes a weak variation of a normal distribution, or a NON uniform distribution. 

We can all agree that it is improbable that we would draw 3 FOW, 3 SHops, or 3 Dark Ritual in an opening hand. We would also agree that it is improbable that there would be 3 Shops, 3 FOW, or 3 Dark Ritual in my top 15 cards.  Can we agree that it is less unlikely – it is less probable than it is probable that there are two Dark Rituals in any set of 15 cards then there is one?  If that is true, then it is a fair inference that the top 7 cards probably do not have a Dark Ritual, a fact made more likely if you have a Dark Ritual in your opening hand. 

I like to think of myself as a rational individual. 

I am quite aware of the fact that the probability of me seeing a Dark Ritual in the top 4 cards should be the same before or after I play Land Grant (assuming the land grant doesn’t thin).  3/53 is 5.66%.  It should be the same either way.  That is the probability that the top card is a Dark Ritual either before or after a shuffle.  The problem is that I don’t believe that that is the right calculation.  Not because I cheated and not because I didn’t randomize.  I believe that randomization isn’t purely and completely random.  This is proven by the simple statement that if the probability of my top card being a multiple is higher than it being a singleton because there are more chances for it to be seen.  Similarly, I think the chance of a Dark Ritual being in the top 8 is slimmer if I have one in my opening hand then it would be if I shuffle.

It’s hard to justify this, but it seems too intuitive to counter.  On average, there should be one Dark Ritual every 15 cards.  If I shuffle, it is like resetting the whole thing.  On average, then, there should be 1 Dark Ritual every 17 cards.  Shuffling will very marginally improve the chance that you’ll see one.  Since the Dark Ritual (on average) for the first 15 had already been seen. 

[Komatteru]

I've more or less decided that traditional probability theory doesn't apply to Magic.  Probability assumes randomness, and Magic decks are anything but random.  Shuffling the cards mixes them up pretty good so you don't know what they are, but they can never be random because your shuffling technique is not perfect.  In addition, how often after a game do you take the lands and toss them into the deck in varying spots so you don't have to shuffle a clump of 6 lands?  That's a variable that cannot be predicted or modeled, which makes the mathematics extremely difficult, to say the least.

[Machinus]

Looking to QM for help in understanding magic theory is a futile endeavor. There is no application of it to this topic.

The probability theory involved here is very clear and is purely mathematical in nature. We can calculate any odds that we want, without difficulty. The only difficulty that arises is with semantics, like in determining what it means to be "cheating" or for something to be "random."

[cssamerican]

If you have ever played on MWS you will understand the difference from random and not-so-random. I have played my Ninja Mask deck in person which has 15 lands, 5 moxen, 1 Black Lotus, and 4 Birds of Paradise  and I RARELY get mana screwed without help; however, on MWS I have played quite a few games where I have two mana sources in my opening hand and never draw another mana source in the next 15 draws. That is because a computer program can get much closer to random than I will allow my deck to get in real life. This is because when I pick up my cards I weave the lands into the deck to ensure they don't get clumped together, the same is true with all the key cards of the deck. I am trying to evenly distribute my cards in the deck everytime I pick them up, while I shuffle them afterwards it is practically impossible to completely remove this stacking. This reduces my chances of getting screwed by bad luck, and makes the deck somewhat more predictable. This is why I believe the strict Mathematic approach is impossible to apply in real life games and many times you have to just make assumptions like you are alluding to; however, that kind of thought will just get you in trouble online because your deck is trully random.

[Machinus]

The chance of drawing at least one Force of Will in your opening hand is 40%.  The chance of there being at least one Force of Will in the subsequent seven cards is also 40%.  So if you drew 14 cards instead of 7, wouldn’t the probability of drawing at least one Force of Will be 80%?

No.

Probability of drawing at least one FoW in 7 cards:      39.949962574466557979%
Probability of drawing at least one FoW in 14 cards:    66.535420960349441693%

This came up from a game scenario that was like this:

Lotus Petal,
Land Grant,
Dark Ritual,
Brainstorm.
3 irrellevant cards

The question is:
Should you Brainstorm before or after you have cast Land Grant, assuming that all the Land Grant does is shuffle.  The question is: if we shuffle, will that increase our odds of seeing another Dark Ritual with the Brainstorm?

Yes, but not because of the shuffle. It will be because the Rituals will then comprise a higher percentage of the deck after you remove a forest.

The real example was this:

Black Lotus,
Lion’s Eye Diamond
Dark Ritual
Land Grant
Darkwater Egg,
Chromatic Sphere,
Brainstorm

With as good as this hand is, I figured that the top of the library is probably cantrips and bad cards.  To see a good mix of cards I’d prefer to shuffle and get the bad cards shuffled into the library.  In other words, I’d say that shuffling the library increases the chances of seeing another Dark Ritual with the Brainstorm.

Is this rational?   That’s what this article seeks to explain.

The answer is the same as before. By removing a forest, the probability of then drawing any non-forest card increases.

...some summary of a few interpretations of QM...

How does this apply to magic?

I really don't think it does.

I think that if you did a million goldfishes after a thousand shuffles for each goldfish, and divided the deck into four piles of 15, you would find a dark ritual in two piles more than 50% of the time.  I think it is likely that a very sizable portion of the time, probably over 40%, you would find a dark ritual in three piles.  Regardless, if we assume that there is one dark ritual in two piles, that is reason enough to shuffle, in my opinion.  Put more risk on it.  What if you had the choice to Spoils of the Vault for Dark Ritual when you already have one in hand.  Your options are to play it before Land Granting (assuming the risk that the land is removed is irrelevant and the thinning is irrelevant), or you can play it after Land Grant.  The point is: you have a Dark Ritual in your opening hand, which decision maximizes your life?

Thinning IS the only relevant action here. There is nothing else going here.

We can all agree that it is improbable that we would draw 3 FOW, 3 SHops, or 3 Dark Ritual in an opening hand. We would also agree that it is improbable that there would be 3 Shops, 3 FOW, or 3 Dark Ritual in my top 15 cards.  Can we agree that it is less unlikely – it is less probable than it is probable that there are two Dark Rituals in any set of 15 cards then there is one?

Probability of opening with three copies of a four-of:                                                                                        0.38040747690383175941%
Probability of there being three of a four-of in the top 15 cards:                                                                        4.1988372450705958350%
Probability of there being 2 rituals in the first 15 cards (or a random selection of 15 cards, they are the same):   21.317173705743025008%

This is mostly obvious.

If that is true, then it is a fair inference that the top 7 cards probably do not have a Dark Ritual, a fact made more likely if you have a Dark Ritual in your opening hand.

All you can infer is what probability tells you:

The probability of there being at least one ritual in your opening 7 cards:                                                                    39.949962574466557979%
The probability of there being at least one ritual in the top 7 cards, after you have drawn 0 in your opening hand:          44.272176214462562964%
The probability of there being at least one ritual in the top 7 cards, after you have drawn 1 in your opening hand:          35.200204900537863912%
The probability of there being at least one ritual in the top 7 cards, after you have drawn 2 in your opening hand:          24.891146589259796807%
The probability of there being at least one ritual in the top 7 cards, after you have drawn 3 in your opening hand:          13.207547169811320755%
The probability of there being at least one ritual in the top 7 cards, after you have drawn 4 in your opening hand:            0.000000000000000000%

I am quite aware of the fact that the probability of me seeing a Dark Ritual in the top 4 cards should be the same before or after I play Land Grant (assuming the land grant doesn’t thin).  3/53 is 5.66%.  It should be the same either way.  That is the probability that the top card is a Dark Ritual either before or after a shuffle.  The problem is that I don’t believe that that is the right calculation.

You are indeed not calculating the probabilities correctly. You need to treat each card draw as an individual event, which requires the use of factorials.

Not because I cheated and not because I didn’t randomize.  I believe that randomization isn’t purely and completely random.  This is proven by the simple statement that if the probability of my top card being a multiple is higher than it being a singleton because there are more chances for it to be seen.  Similarly, I think the chance of a Dark Ritual being in the top 8 is slimmer if I have one in my opening hand then it would be if I shuffle.

It’s hard to justify this, but it seems too intuitive to counter.  On average, there should be one Dark Ritual every 15 cards.  If I shuffle, it is like resetting the whole thing.  On average, then, there should be 1 Dark Ritual every 17 cards.  Shuffling will very marginally improve the chance that you’ll see one.  Since the Dark Ritual (on average) for the first 15 had already been seen.

Your intuition is misleading you here. Shuffling has absolutely no effect on the probabilities. Only deck thinning does. You are making a similar mistake as you made before when you "inferred" that your opponent did not have the nuts game 3 because they did in games one and two. If you do not measure the probability of there being a ritual in the top 15 cards, then all you know is the probability that there is one - and this does not change no matter how many times you shuffle your deck.

[Ishi]

It's not a coincidence that the article's logic is hard to follow.  If you clean up your argument and put it in a simple form like A => B => C => you should cast that land grant to shuffle, I think logical errors will jump out immediately.

1 problem is you seem to be confusing two probabilities:

* The chance you will see 2 dark rituals in a 15 card pile
* The chance of the same when you know that there is 1 ritual in the first 8 cards.

EDIT:

Also, uniform distribution doesn't mean "there is no pattern" (which isn't a very precise statement), it means each card is equally likely to be in all deck positions.  You will definitely see patterns, for example after the 4th ritual there will never be another ritual in the deck.  Can you give a link to "even distribution", I couldn't find it on google.

[Machinus]

If you have ever played on MWS you will understand the difference from random and not-so-random. I have played my Ninja Mask deck in person which has 15 lands, 5 moxen, 1 Black Lotus, and 4 Birds of Paradise  and I RARELY get mana screwed without help; however, on MWS I have played quite a few games where I have two mana sources in my opening hand and never draw another mana source in the next 15 draws. That is because a computer program can get much closer to random than I will allow my deck to get in real life. This is because when I pick up my cards I weave the lands into the deck to ensure they don't get clumped together, the same is true with all the key cards of the deck. I am trying to evenly distribute my cards in the deck everytime I pick them up, while I shuffle them afterwards it is practically impossible to completely remove this stacking. This reduces my chances of getting screwed by bad luck, and makes the deck somewhat more predictable. This is why I believe the strict Mathematic approach is impossible to apply in real life games and many times you have to just make assumptions like you are alluding to; however, that kind of thought will just get you in trouble online because your deck is trully random.

In real life, your deck is more evenly distributed. On MODO and MWS, the deck is much closer to a random distrubution, as you said. This does change what cards you draw, but it doesn't change the probabilities, which means you shouldn't play any differently. A good player will play correctly (that is, according to probability) and not let themselves be fooled by their intuition, or even by a possibly skewed set of testing data.

[Godder]

The probability of NOT drawing a FOW in any hand of 7 is 60%.  60% * 60% = 36%.  The chance of NOT seeing a FOW in the top 14 cards is 36%. In other words, the chance of seeing at least one FOW in the top 14 cards is 74%, which for reasons already explained, is less than the 80% which one would obtain by adding the probabilities of seeing at least one FOW from the first and second set of 7 cards.

Minor arithmetic error .

[J J P]

I think you missed the point of Schrödinger's cat. The intend of this hypothetical experiment is to show that quantom theory in it's current form can't be applied to macroscopical systems because it leads to results that don't represent reality. Basing conclusions on something paradox is strange (and mathematically nonsense).

[Elric]

Here’s how I’d say it:

Assume that your deck is in random order (all possible orders of cards are equally likely): If you have a 60 card deck with 4 Dark Rituals and you draw a Dark Ritual on the first card then your chance to draw a Dark Ritual on the second card is the chance to draw a Dark Ritual from a randomized 59 card deck with 3 Dark Rituals.

On the other hand, if your deck isn’t in random order and Dark Rituals are “more spread out� across the deck than they should be if all orderings were equally likely, then the chance to draw a Dark Ritual on the second card won’t be the same as the chance to draw a Dark Ritual from a randomized 59 card deck with 3 Dark Rituals (in this case, since the Rituals are “more spread out� it’s going to be lower). 

If you mana-weave your deck before shuffling, the amount of shuffling that you do probably won’t randomize the deck to the point when it is in completely random order (it will have less clumps than it should).  With that said, if this “lack of clumping� effect is significant enough to cause a major change in playstyle or deck construction, you’re probably cheating.  The point of shuffling is to randomize the deck.

In a randomized deck of finite size (with 1 or more Dark Rituals), your chance to get 2 Dark Rituals on your first two cards is less than the square of the chance to get a Dark Ritual on the first card (i.e., your chance to get a Ritual on the second card, given that you already got a Ritual on the first card, is less than your chance of getting the initial Ritual was).

Imagine, though, that instead of a deck with 4 Dark Rituals in 60 cards you have a randomized deck with an infinite number of cards, 1/15th of which are Dark Rituals.  Then the chance to get 2 Dark Rituals on the first two cards equals the square of the chance to get a Dark Ritual on the first card (the chance to get a Dark Ritual is the same on each card, regardless of what cards have already been drawn).  This isn’t the case in with decks in real life, though.

[Whatever Works]

I have to agree with Smennen. Whether people like it or not... EVERYTHING can be applied to statistics...  Magic is no acception.

This is one reason why I get ennoyed when people say... Oh you got lucky... However, if you define "luck" as achieving a draw (such as topdecking a black lotus) that would be approximatly 3% then you can describe "luck" as meeting a very achievable goal that is low percentage...Then that is acceptable...

The thought that the system is flawed by inperfect shuffling is to me fairly irrelevent. Non "perfect" shuffling does change odds as stated, but only very very little. Nit-picking 1-2% doesnt matter as long as your not aware where your cards are.

Understanding statistics is very very good thing to have playing magic. The best players will take advantage of it, and will do very minor things to try and gain a better understanding of the game state. An example would be counting the # of FoW and Mana drains in a graveyard to calculate how many are left, and the odds that a player has 1...

Kyle L

[Komatteru]

Non "perfect" shuffling does change odds as stated, but only very very little. Nit-picking 1-2% doesnt matter as long as your not aware where your cards are.

You really don't understand how bad most shuffling really is.  It changes the order maybe 30% of the cards in the deck.  "Not aware" of where your cards are doesn't cut it.  The mathematics assumes that the object in question is random.  That's the hypothesis.  In mathematics, if you don't meet the hypothesis, the theorems don't apply.  That's where we are here.  You can try it, but the mathematics most people use just isn't complex enough to deal with all the unknown variables variables and such that occur.  You can describe it more accurately using stochastic models, but those are kinda a pain.

[Machinus]

Non "perfect" shuffling does change odds as stated, but only very very little. Nit-picking 1-2% doesnt matter as long as your not aware where your cards are.

You really don't understand how bad most shuffling really is.  It changes the order maybe 30% of the cards in the deck.  "Not aware" of where your cards are doesn't cut it.  The mathematics assumes that the object in question is random.  That's the hypothesis.  In mathematics, if you don't meet the hypothesis, the theorems don't apply.  That's where we are here.  You can try it, but the mathematics most people use just isn't complex enough to deal with all the unknown variables variables and such that occur.  You can describe it more accurately using stochastic models, but those are kinda a pain.

Sure, that's fine, but there's no way you can make any predictions based on the non-random distribution of real decks. It's preposterous to propose that you could somehow model this. At best it's a player's experience, and at worst its just totally wrong and worse than using standard probability.

[PipOC]

It changes the order maybe 30% of the cards in the deck.

That's funny because a single "perfect" crush shuffle changes the position of every card in the deck, so could you explain what you mean by that?

[Vegeta2711]

I think I speak for everyone here when I ask:  Is randomness as random as the randoms who play the random?

[Whatever Works]

Non "perfect" shuffling does change odds as stated, but only very very little. Nit-picking 1-2% doesnt matter as long as your not aware where your cards are.

You really don't understand how bad most shuffling really is.  It changes the order maybe 30% of the cards in the deck.  "Not aware" of where your cards are doesn't cut it.  The mathematics assumes that the object in question is random.  That's the hypothesis.  In mathematics, if you don't meet the hypothesis, the theorems don't apply.  That's where we are here.  You can try it, but the mathematics most people use just isn't complex enough to deal with all the unknown variables variables and such that occur.  You can describe it more accurately using stochastic models, but those are kinda a pain.

Sure, that's fine, but there's no way you can make any predictions based on the non-random distribution of real decks. It's preposterous to propose that you could somehow model this. At best it's a player's experience, and at worst its just totally wrong and worse than using standard probability.

It still comes down to appliable situations regardless if every single card is perfectly random... Example...

You have 3 FoW's in your graveyard... 1 left in your deck... Yor library in 13 cards... You cast brainstorm... Regardless of the accuracy of your shuffling... Its still going to see 3 of 13 cards... leaving about a 30% chance... I know its not 30%... I know the cards were not perfectly random... Yet, I have a good sense of my chances...

REMEMBER: People play poker... and there is no way in hell you can get me to believe that the poker decks are perfectly randomized either... I dont see the dealers riffle shuffling 8 times almost ever... Usually 4-6 sinse people like a fast moving table... If proffessional poker players can apply statistics to that card game why cant we?

Statistics arent exact... The word Stat general refors a sample... the deck being the same... stats have room for more error then a population that takes all things into account... (but this is hardly the point)...

Almost nothing has perfect randomness... there is almost always bias... that doesnt prevent statistics from being an affective form of not precise problem solving, but as a tool of reasonability, and likely occurences...

Kyle L

[Machinus]

Non "perfect" shuffling does change odds as stated, but only very very little. Nit-picking 1-2% doesnt matter as long as your not aware where your cards are.

You really don't understand how bad most shuffling really is.  It changes the order maybe 30% of the cards in the deck.  "Not aware" of where your cards are doesn't cut it.  The mathematics assumes that the object in question is random.  That's the hypothesis.  In mathematics, if you don't meet the hypothesis, the theorems don't apply.  That's where we are here.  You can try it, but the mathematics most people use just isn't complex enough to deal with all the unknown variables variables and such that occur.  You can describe it more accurately using stochastic models, but those are kinda a pain.

Sure, that's fine, but there's no way you can make any predictions based on the non-random distribution of real decks. It's preposterous to propose that you could somehow model this. At best it's a player's experience, and at worst its just totally wrong and worse than using standard probability.

It still comes down to appliable situations regardless if every single card is perfectly random... Example...

Statistics arent exact... The word Stat general refors a sample... the deck being the same... stats have room for more error then a population that takes all things into account... (but this is hardly the point)...

Almost nothing has perfect randomness... there is almost always bias... that doesnt prevent statistics from being an affective form of not precise problem solving, but as a tool of reasonability, and likely occurences...

Kyle L

I think perhaps you are misunderstanding me. The probability I utilized in my response to the first post is based on a random deck distrubution. This is still a GOOD model for the even distribution (non-random) of real world decks, and it is still the wisest metric that a good player can use to predict future draws. However, there is likely to be a greater standard deviation than for an average sample of truly randomized decks. In addition, there is no possible way to improve the mathematical model, that is already being used, to account for the nonrandom distrubution effect of normal shuffling. Furthermore, there is no way to specifcy exactly how far off it is. It is still the best available model, even if it is does not perfectly predict the probabilities.

[cssamerican]

Just so people don't take my comments the wrong way let me define a term before I start replying. When I refer to a deck being stacked I don't mean completely stacked with some planed distribution, but rather I pick up my four Illusionary Mask and weave them into the deck at different points to ensure they don't start off clumped together.

In real life, your deck is more evenly distributed. On MODO and MWS, the deck is much closer to a random distrubution, as you said. This does change what cards you draw, but it doesn't change the probabilities, which means you shouldn't play any differently. A good player will play correctly (that is, according to probability) and not let themselves be fooled by their intuition, or even by a possibly skewed set of testing data.

That isn't necessarily true. For example, if I play Spoils of the Vault turn 1 for a 4 of card in a completely random deck 14% of the time it will result in a suicide; however, that percentage will be lower if the deck had been stacked prior to shuffling. This is the most obvious and direct way to illustrate the real difference between a random deck and one that isn't so random. I could think of a lot of other scenarios where knowing that the odds of cards being clumped together are reduced would effect my decisions in an actual game. I think this is what Steve has perceived through experience which to a degree is actually true, though hard to prove mathematically.

If you mana-weave your deck before shuffling, the amount of shuffling that you do probably won’t randomize the deck to the point when it is in completely random order (it will have less clumps than it should).  With that said, if this “lack of clumping� effect is significant enough to cause a major change in playstyle or deck construction, you’re probably cheating.  The point of shuffling is to randomize the deck.

First, I don't think you could totally randomize a deck in a reasonbale amount of time. You can shuffle to the point where no card is in the same position, but that still doesn't necessarily mean the distribution of cards has been changed a significant amount to be considered completely random. This doesn't mean I'll know with any certainty what my next draw is, that would be cheating. However, if you weaved a set of cards into your deck prior to shuffling you have some information (It is less likely those cards will be clumped together), and you should use all the information you can when making decisions

[Elric]

If you mana-weave your deck before shuffling, the amount of shuffling that you do probably won’t randomize the deck to the point when it is in completely random order (it will have less clumps than it should).  With that said, if this “lack of clumping� effect is significant enough to cause a major change in playstyle or deck construction, you’re probably cheating.  The point of shuffling is to randomize the deck.

First, I don't think you could totally randomize a deck in a reasonbale amount of time. You can shuffle to the point where no card is in the same position, but that still doesn't necessarily mean the distribution of cards has been changed a significant amount to be considered completely random. This doesn't mean I'll know with any certainty what my next draw is, that would be cheating. However, if you weaved a set of cards into your deck prior to shuffling you have some information (It is less likely those cards will be clumped together), and you should use all the information you can when making decisions

If this isn't cheating it should be.  Here's my opinion of what should be cheating: If you have 24 lands and you make sure that your deck has 2 lands in every 5 cards, that's cheating.  If you put your Fastbond next to your Crucible of Worlds, that's cheating.  If you look at which cards are next to each other while searching with a fetchland and then don't really shuffle well, that's cheating.  If you "reweave" your deck to have less mana clumps while searching with a fetchland and don't reshuffle well, that's cheating. 

People may not shuffle perfectly but gaming the system to your advantage should be cheating.  It's cheating that you're unlikely to catch- I know a guy who could probably stack most of a deck when you thought he was shuffling randomly- and it's often minor, but that doesn't mean it isn't cheating.  When you have a limited amount of time to shuffle and the cards start stacked non-randomly, you can’t totally avoid some non-random element in the result. 

If you build your deck on the assumption that you are going to be the beneficiary of significantly "less clumped than random� mana distributions then you're building your deck around the assumption that you're going to be able to cheat (possibly with the aid of an opponent who doesn't shuffle enough or have time to shuffle enough).  As far as I know there’s no rule that say “it’s ok to stack your deck as long as your opponent isn’t clever enough to stop it or catch you.�

If you mark all of your lands with a dot so that you can tell when they’re on the top of your deck, this is cheating.  If you mark 2/3 of your lands and 1/3 of your non-lands so that you can tell only probabilities by looking at the mark on the top card, this is still cheating even though you haven’t cheated to the point when you know what the top card will definitely be based on the marking. 

Likewise, if your deck is closer to alternating mana/non-mana than it should be then whenever you draw a card you are given information about the next card that you shouldn't have.  You also have a more favorable distribution of cards in general, even if it isn't stacked to specific sets of cards next to each other.

[Komatteru]

Sure, that's fine, but there's no way you can make any predictions based on the non-random distribution of real decks. It's preposterous to propose that you could somehow model this. At best it's a player's experience, and at worst its just totally wrong and worse than using standard probability.

Clearly, you haven't delved deep enough into mathematics.  Look up "stochastic processes" and see what those are all about.  I know very little about it myself, but you'd be amazed at what you can model.  How accurate your model is depends on what variables you want to ignore.

[Elric]

Sure, that's fine, but there's no way you can make any predictions based on the non-random distribution of real decks. It's preposterous to propose that you could somehow model this. At best it's a player's experience, and at worst its just totally wrong and worse than using standard probability.

Clearly, you haven't delved deep enough into mathematics.  Look up "stochastic processes" and see what those are all about.  I know very little about it myself, but you'd be amazed at what you can model.  How accurate your model is depends on what variables you want to ignore.

I think what Machinus means is that there's no way to come up with the correct assumptions for a non-random distribution of decks.  What are you going to assume?  That mana is less than randomly clumped?  That each 4-of (or 3-of or 2-of) card is less than randomly clumped?  That Fastbond and Crucible are more likely to show up next to each other?  That all of these are true? 

Just assume that you stack your deck and then use some specific non-randomizing means of shuffling?  Assume that the deck is more likely to be in sorted alphabetic order, or that cards with the same artist will clump together?  It isn't a question of math- it's a question of trying to know something that essentially can't be known to any degree of accuracy.  If people aren't cheating, randomness is a pretty good assumption.

[Machinus]

Sure, that's fine, but there's no way you can make any predictions based on the non-random distribution of real decks. It's preposterous to propose that you could somehow model this. At best it's a player's experience, and at worst its just totally wrong and worse than using standard probability.

Clearly, you haven't delved deep enough into mathematics.  Look up "stochastic processes" and see what those are all about.  I know very little about it myself, but you'd be amazed at what you can model.  How accurate your model is depends on what variables you want to ignore.

That is still based on randomness. The point of contention here is that real-world shuffling is not random. The problem with trying to adjust the probability predictions is that there is no way to know the specific character of the de-randomization. If we could understand this, then you could shuffle a specific way in order to determine how your cards were organized, but the best we can do is "randomize" the deck. You could say that pile and riffle-shuffling is roughly going to distribute the cards more evenly throughout the deck, and therefore have some effect on the probability of drawing cards based on what is already drawn. To some extent this phenomena can be exploited by experienced magic players who shuffle well and who are very familiar with their decks. However, your shuffling could easily redistrubute the cards in a different manner, or in an uneven way. The standard probabilitistic arguments then provide a good basis for making predictions.

While I agree that looking at a more even distrubution function makes sense based on good shuffling techniques, I think any preference towards this line of thought is mostly due to fallacious intuition, and that the statistical deviation is probably very small as well as very complicated to describe. Don't you think that following probabilistic arguments is the best choice a player can make?

[Komatteru]

The standard probabilitistic arguments then provide a good basis for making predictions.

This doesn't follow at all from what you said earlier.  Essentially, your argument here goes: We can either jump off a bridge or not jump off the bridge.  There problems in jumping off the bridge, but we cannot accurately know what will happen if we don't jump.  We could be run over by a bus, attack by wombats, etc.  However, if we jump, we know what will happen, so it provides us with a good basis for what we might do.  Therefore, we should jump. 

Notice the holes there.

Don't you think that following probabilistic arguments is the best choice a player can make?

Actually, I think statistics are completely worthless, but that's just me. 

The trick is not to get involved with numbers.  Either an event is "likely" or it's not.  People get into "OMG!! I had a 35.64767% chance to win that game because it all relied on my opponent not having card X."  That's worthless.

Math never works in the real world like it should.  I know this quite well.

[Elric]

The standard probabilitistic arguments then provide a good basis for making predictions.

This doesn't follow at all from what you said earlier.  Essentially, your argument here goes: We can either jump off a bridge or not jump off the bridge.  There problems in jumping off the bridge, but we cannot accurately know what will happen if we don't jump.  We could be run over by a bus, attack by wombats, etc.  However, if we jump, we know what will happen, so it provides us with a good basis for what we might do.  Therefore, we should jump. 

Notice the holes there.

You have to add the all-important "And we know that jumping and not jumping will be quite similar" (since we know that decks will be pretty close to random if people aren't cheating).  An approximation in an assumption is good if your result would still be a good predictor of what actually happens if that approximation was off by the amount we'd expect it to be off by in real life.

That's why assuming random decks is fine and assuming players have psychic powers isn't fine. 

[Machinus]

The standard probabilitistic arguments then provide a good basis for making predictions.

This doesn't follow at all from what you said earlier.  Essentially, your argument here goes: We can either jump off a bridge or not jump off the bridge.  There problems in jumping off the bridge, but we cannot accurately know what will happen if we don't jump.  We could be run over by a bus, attack by wombats, etc.  However, if we jump, we know what will happen, so it provides us with a good basis for what we might do.  Therefore, we should jump.

Huh? My post was pretty clear, and devoid of logical errors. Here is what I said:

While I agree that looking at a more even distrubution function makes sense based on good shuffling techniques, I think any preference towards this line of thought is mostly due to fallacious intuition, and that the statistical deviation is probably very small as well as very complicated to describe.

Here is a shorter version: "assuming the deck is truly randomized is a good approximation." That's the same thing I said before, and afterwards. I don't see any holes in my argument.

[benthetenor]

The standard probabilitistic arguments then provide a good basis for making predictions.

This doesn't follow at all from what you said earlier.  Essentially, your argument here goes: We can either jump off a bridge or not jump off the bridge.  There problems in jumping off the bridge, but we cannot accurately know what will happen if we don't jump.  We could be run over by a bus, attack by wombats, etc.  However, if we jump, we know what will happen, so it provides us with a good basis for what we might do.  Therefore, we should jump. 

Notice the holes there.

Don't you think that following probabilistic arguments is the best choice a player can make?

Actually, I think statistics are completely worthless, but that's just me. 

The trick is not to get involved with numbers.  Either an event is "likely" or it's not.  People get into "OMG!! I had a 35.64767% chance to win that game because it all relied on my opponent not having card X."  That's worthless.

Math never works in the real world like it should.  I know this quite well.

I agree. Blindly following statistics in Magic is equivalent to playing half of the game. What do I care what the probability is that (s)he drew that one card, or even the probability of drawing a certain card is? It's not particularly changeable within a game, and if you're playing/deckbuilding correctly then you're already maximizing the number of cards you see on any given turn.

It's just like in poker. You can't go by the probability of someone being dealt Aces because you'd never, ever expect it since it only happens (statistically) about 0.45% of the time, or one time in 220 hands. Do you just sit out every 220th hand? No, you play according to the probability that your opponent was dealt Aces based on his body language/betting pattern/nervous tic/whatever and adjust your game accordingly. Every card game is based less on probability than you would think. This is of course because statistics is based upon running a simulation hundreds of thousands of times, whereas in any card game the simulation can only be run once a game. In poker the odds matter, as poker is really a matter of assigning an amount of chips to invest in any given hand, basing it on the odds of winning or not. Better odds, bet more. That's why when you're getting pot-odds of 4.7 to 1, it's sometimes appropriate to call an all in with the dreaded 4-6 offsuit. In Magic, you either win or you lose, and you can't control how much that win or loss effects your tournament life because you can't wager more or less than a single game on any given game.

In more practical terms, knowing the odds can effect your play, but the human being isn't a calculator. In poker the odds for any given game are widely known because the deck never changes. Even then, only the very best poker players, having played millions of hands with the same deck, know the probabilities to more decimal places than the average player. In Magic, decks vary by as much as an entire 60 cards and as little as a single card or not at all. Rather than having 13 different kinds of cards in 4 different suits, you could theoretically have 60 different cards, or cards in differing configurations (such as not all as 4-ofs). Calculating odds on the fly by hand for any given deck would take 10 minutes off the opening 7 and an additional 10 minutes for every card drawn after that as probabilities are adjusted. If you were good, you'd get it down to 7. Even then, you'd get about 6 minutes into the tournament and then get DQ'ed for stalling.

In the end, studying the odds of Magic would take a lifetime, at which point you'd know barely more than the average n00b. If you draw a card, it's less likely that you'll draw another copy of that card anytime soon unless you find a way to draw more cards/tutor for a card/use card selection. The odds of drawing a FOW in the opening 7 doesn't really matter because you either will or you won't, you can't tweak the deck anymore than having 4 FoW to increase those probabilities, and knowing that you should have it in your opening hand 4 games in 10 doesn't prevent you from never getting it in your opening hand or from having it in your opening hand every single game. Figure the odds if you want to or if it interests you, but it's just not a productive path of study.

[Elric]

benthetenor- do you see anyone advocating "blindly following the stats"?  I don't think anyone says that knowing the exact probability of being dealt aces is the entire game.  I think this is mainly a straw man argument.

However, you certainly want to know more than "likely/unlikely" as JDizzle suggested.  0.5%, 5%, and 40% are all events that happen less than half of the time, but there's a big difference along this range.

Lastly, this ties in with the entire approximation to a random distribution.  Even if magic cards are very slightly more likely to be in some arrangement than others (due to decks starting out in particular orders before shuffling + imperfect shuffling), you don't care about the exact values you get enough for it to matter- you just want a fairly accurate idea what's going on.

[benthetenor]

benthetenor- do you see anyone advocating "blindly following the stats"?  I don't think anyone says that knowing the exact probability of being dealt aces is the entire game.  I think this is mainly a straw man argument.

However, you certainly want to know more than "likely/unlikely" as JDizzle suggested.  0.5%, 5%, and 40% are all events that happen less than half of the time, but there's a big difference along this range.

Lastly, this ties in with the entire approximation to a random distribution.  Even if magic cards are very slightly more likely to be in some arrangement than others (due to decks starting out in particular orders before shuffling + imperfect shuffling), you don't care about the exact values you get enough for it to matter- you just want a fairly accurate idea what's going on.

Did you read my post? What's the point of caring about the odds if you can't adjust your strategy to use them to your advantage? All I wish to know is likely v. unlikely. If a play is likely not going to work, then I'll avoid it. If you want to take the time to discover the odds of all of the relevant plays, you will be able to find the statistically "best" play. Given that, you've just spent 5 to 10 minutes thinking about all of the possible plays at one given instant when you could have arrived at the same conclusion in 15 seconds by simply knowing which play is likely to succeed and which is likely to fail and going with that.

If you know that you're going to lose 5% more with one play than another over the course of one game that's virtually statistically insignificant. Given the fact that most players do not play the exact same deck over any two tournaments, and given all the millions of possible plays that arise from any one iteration of a deck, a 5% difference arising from one of the millions of plays simply does not matter. Changing the deck by a single card will cause you to need to recalculate all of those probabilities that you've spent a lifetime computing, and yet for all that work, you'd be lucky to even make the same play twice over the course of a tournament. In most tournaments, the 5% difference will never surface, even if you had to make the exact same play in the exact same gamestate every game. This is of course because a tournament isn't an infinite population. In Type I, it's usually a maximum population of about 33 or so games, depending on how well you do. In an infinite population, 5% is a big difference. In a population of 33 games with infinite possible gamestates, the difference WILL NEVER matter. Knowing which play will probably work and which one will probably not, however, is something that will prove important over the course of a tournament, and is really all that one can hope to glean from even a year's worth of playtesting.

[lordmayhem]

It changes the order maybe 30% of the cards in the deck.

That's funny because a single "perfect" crush shuffle changes the position of every card in the deck, so could you explain what you mean by that?

Changing the position of every card in the deck is not equal to randomization. Say you have 10 cards left in your library.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Typically you would grab cards 6-10 and crush them into cards 1-5 achieving:

1, 6, 2, 7, 3, 8, 4, 9, 5, 10.

Typically you would then grab 8, 4, 9, 5 and 10 and crush them into 1, 6, 2, 7 and 3, thus achieving:

1, 8, 6, 4, 2, 9, 7, 5, 3, 10.

Even though the shifting of card positions is not as easy to predict as that, since there will be times were 2-3 cards go between 2 other cards, randomization is vastly different. I produced a simple program in C# to generate random numbers from 1 to 10, ensuring that the number wasn't already created and appending the number to an array. The results I got were as follows:

6, 1, 3, 10, 4, 9, 2, 8, 7, 5,
6, 1, 2, 7, 10, 8, 9, 3, 5, 4,
3, 6, 9, 2, 8, 4, 5, 1, 7, 10,
10, 1, 9, 4, 5, 2, 6, 7, 3, 8,
9, 10, 1, 8, 3, 5, 6, 2, 4, 7,
2, 7, 5, 9, 6, 8, 4, 3, 10, 1,

On attempting a randomization from 1 to 60, the results were:

22, 4, 55, 51, 27, 3, 50, 48, 19, 54, 58, 39, 46, 52, 28, 15, 47, 20, 41, 35, 6, 59, 11, 31, 57, 18, 12, 21, 25, 1, 10, 40, 37, 42, 44, 60, 26, 7, 36, 16, 30, 53, 2, 24, 14, 34, 33, 43, 32, 13, 38, 23, 45, 49, 56, 29, 5, 9, 8, 17,

20, 9, 41, 36, 29, 53, 43, 17, 25, 50, 6, 46, 40, 56, 14, 47, 2, 10, 27, 59, 23, 57, 35, 8, 55, 32, 7, 24, 28, 30, 4, 42, 19, 5, 3, 38, 1, 18, 45, 39, 54, 34, 37, 12, 33, 58, 22, 11, 26, 31, 16, 21, 13, 60, 49, 15, 44, 51, 48, 52,

11, 39, 37, 33, 19, 41, 25, 14, 6, 32, 60, 12, 48, 53, 27, 58, 1, 45, 9, 30, 22, 20, 36, 23, 59, 47, 49, 52, 29, 50, 2, 24, 51, 56, 44, 42, 16, 26, 46, 38, 5, 17, 31, 43, 4, 40, 57, 55, 8, 13, 15, 28, 10, 3, 21, 34, 7, 54, 18, 35,

Which is what causes the major difference between crush shuffling, which is just a shifting of cards (albeit in an order unknown to us), and the random shuffling that a computer generates on MWS or Apprentice.

EDIT : Damn. So many numbers  :shock:

EDIT EDIT : To get it to look more like Magic than just number I got it to display some card names instead of some numbers. I give you : Randomized Sui .... and numbers...

48, 58, Withered Wretch, Swamp, 56, Swamp, Hypnotic Specter, Wasteland, 54, Swamp, Sinkhole, Dark Ritual, 57, Dark Ritual, Swamp, Swamp, Dark Ritual, Wasteland, Swamp, Sinkhole, Swamp, Hypnotic Specter, 50, Withered Wretch, 44, Swamp, 55, Sinkhole, Hypnotic Specter, Phyrexian Negator, Sinkhole, Swamp, Swamp, Phyrexian Negator, 53, Withered Wretch, 49, 47, 60, Dark Ritual, Phyrexian Negator, 41, 45, Phyrexian Negator, 46, Withered Wretch, 42, Swamp, Swamp, 52, Swamp, Swamp, Swamp, 59, 51, Swamp, Hypnotic Specter, Wasteland, Wasteland, 43,

Swamp, 49, Swamp, 59, Swamp, 42, Swamp, 53, Phyrexian Negator, Hypnotic Specter, Sinkhole, Dark Ritual, Swamp, Dark Ritual, 44, Swamp, Swamp, 45, Swamp, 50, Dark Ritual, Phyrexian Negator, Hypnotic Specter, Wasteland, 56, 46, Wasteland, Hypnotic Specter, 60, 52, Swamp, Swamp, Sinkhole, Swamp, Wasteland, Withered Wretch, 47, Dark Ritual, Withered Wretch, 57, Swamp, Swamp, 58, Wasteland, Phyrexian Negator, 43, Swamp, 41, 48, Phyrexian Negator, Withered Wretch, Withered Wretch, Swamp, Sinkhole, Swamp, Sinkhole, Hypnotic Specter, 54, 51, 55,

Hypnotic Specter, Swamp, 56, 59, Wasteland, Phyrexian Negator, 60, Phyrexian Negator, Hypnotic Specter, Swamp, Phyrexian Negator, Wasteland, 53, Sinkhole, Swamp, 47, 54, Wasteland, Swamp, Swamp, Wasteland, Sinkhole, Hypnotic Specter, Swamp, 46, Withered Wretch, Swamp, Swamp, 42, Swamp, Withered Wretch, 43, Sinkhole, Swamp, Dark Ritual, 49, 51, Withered Wretch, 41, Dark Ritual, Dark Ritual, Swamp, Swamp, 57, 44, 50, 55, 52, Phyrexian Negator, 48, Swamp, 45, Swamp, Swamp, Hypnotic Specter, Sinkhole, Withered Wretch, 58, Dark Ritual, Swamp,

[Elric]

Did you read my post? What's the point of caring about the odds if you can't adjust your strategy to use them to your advantage? All I wish to know is likely v. unlikely. If a play is likely not going to work, then I'll avoid it. If you want to take the time to discover the odds of all of the relevant plays, you will be able to find the statistically "best" play. Given that, you've just spent 5 to 10 minutes thinking about all of the possible plays at one given instant when you could have arrived at the same conclusion in 15 seconds by simply knowing which play is likely to succeed and which is likely to fail and going with that.

Huh?  You can adjust your strategy to use the odds to your advantage.  It's just that knowing the exact odds of your opponent having certain cards down to the last % doesn't change much about your strategy compared to knowing approximate odds.  For example, knowing that your opponent having a turn 1 Oath happens 35% of the time vs. 15% of the time vs. 3% of the time will make a big difference.  18%, 17%, or 16%- not so much.

I don't know if you mean that people will have an intuitive sense of this probability that is very accurate already, but I think this is probably not the case in general.  Of course, math can't account for opponent's mulligan process, etc., and that's why no one says that you should only use math.