Fun with Math: Probabilities

[Harlequin]

One thing I find interesting is that Probability is as theoretical as say the exsistance of God.  Every Probability Model is based on an assumed distribution.  For example, every probability problem has to start with an explicit "fair" coin.   Now we can extrapolate that probably distribution: 50% True, and 50% False out until our minds explode.  But in reality, its impossible to ever know if a coin is indeed fair. 

What we can know in reality is the Statistics of an experiment.  So for example, we flip a coin 1000 times and come up with 501 Heads, and 499 Tails.  This is a statistic, and we can derive information from it.  But ultimatly we use Probability to aid statistics in defining probability!  So for example we can say that given our experiment there is a 99% probability that the probability distribution of the coin is 50:50.  We can also say that there is a 0.01% chance that the probability distribution is 99% chance of heads, and 1% chance of tails.  The great part is that probability tells us that we cannot simply throw away our 99:1 model because there is a chance that Tails got 'lucky' in our experiment. 

The Probability exsist because of observable Statistic we find in nature.  However Statistics requires Probability in order to be determinate.  While the two concepts are fundamentally opposite, they are also interconnected.  This is somewhat the same in Magic.  Probability can be useful, as well as Statistics (or experiance to put a differant name on it).  And neither is useful without the other.


Luck is an entirely differant animal.  If Probabilty = God, Statistic = Reality, then Luck = Prayer (or worship).   Luck is how we explain (and cope with) Statistics we experiance that differ from Probability we know ... much in the same way we use Prayer to understand the differances in our Reality and our perception of God. 

The danger is putting too much importance in Luck (think of it like putting too much faith in prayer).  Saying that "He is Lucky because he had Force of Will for Every one of my Threats."  Is a dangerous statement.  Try and concider that perhapse he had Force of Will at those moments because he was skilled enough to know when to hold back.

[arctic79]

Now let's think about this coin flipping for a second.  Heads wins 501 times while tails wins 499 times, so pretty much 50:50.  What isn't factored in is the chance (luck, probability) that you will flip tails 25 consecutive times in that trial instead of 13 heads and 12 tails.  You can calculate that there is a certain percentage of times this will occur. But you can't calculate when this will occur, just the probabilty of it happening.

I don't think that anyone of skill and intelligence (unless they are superstitious) would say that his opponent was lucky for having Force of Will at the right time.  What you can say is your opponent was lucky that their only out was top decked at the right time, or you can say that you made a bad choice earlier in the game that allowed that spell to resolve.  I don't care who you are, you will always chalk up the top decked only possible win to luck.

[SiegeX]

Now let's think about this coin flipping for a second.  Heads wins 501 times while tails wins 499 times, so pretty much 50:50.  What isn't factored in is the chance (luck, probability) that you will flip tails 25 consecutive times in that trial instead of 13 heads and 12 tails. 

As it shouldn't.  You can flip a coin and have land up heads 10,000 times and tails 0 times. However, the chance of flipping tails on the next flip is still 50:50.  We can only look back and think wow, that was a really unlikely streak I had there, but the past does not dictate future flips as they are (statistically) independent of each other.

To be honest, I'm not really sure of the point you were trying to convey.

On a personal note, I really enjoy this discussion.

[arctic79]

The point I was trying to make was that as unlikely as the scenario is to occur, what would you chalk it up to Luck or Probability?  We know it is probable, we can calculate that it will happen, yet when faced with it in a real world application the first words out of the majority of the population is Luck. 

So I think it is safe to say that when we say luck we actually mean probable. Tomato, Tomatoe.
The danger which was pointed out earlier is when someone actually believes in luck and discounts probability.
For giggles I just looked up various definitions of luck and probability.  They are essentially the same excpect that probability refers to the occurence of events and luck refers to the occurence of personal events.  So when we are talking about Magic where we have both elements in play (the deck and the player), we could theoretically conclude that we can draw card "A" a certain percentage of the time, but dependant on the individual that percentage for drawing card "A" is higher or lower.

I'm not saying this is solid fact, I just find this a good discussion for getting the old brain juices flowing and quite enjoyable reading other peoples takes on this subject.

[Zieby]

I see it this way and I hope it is correct English

What is my defenition of "Luck"[/B] in the game Magic?
A chance that unfolds in a strategic advantage, although the options at success where low.

Greetz Arjan

[Laplie]

As it shouldn't.  You can flip a coin and have land up heads 10,000 times and tails 0 times. However, the chance of flipping tails on the next flip is still 50:50.  We can only look back and think wow, that was a really unlikely streak I had there, but the past does not dictate future flips as they are (statistically) independent of each other.

Taking a side detour for a second:
If a coin flips up heads 10,000 times in a row, you better believe I'm going to not think the chance of flipping tails on the next flip is 50:50.  The 50:50 figure is based on the assupmtion of a fair coin.  The fair coin is like the "frictionless plane" in physics or the "0 resistance wire" in electronics.  It's decent for modeling, but does not exist in reality.  You can use statistics to find the likelyhood that the coin is fair, and in this case that likelyhood is very low (though still possible).

If a coin flips heads 10,000 times in a row, I begin to question the fairness of the coin and the fairness of the flip.  For me, the same applies to Magic.  If someone has their unlikely 7 card combo in the opening hand of every game I play vs them, I begin to question the legality of their deck and whether or not their shuffle leaves their deck random state or not (cheating or incompetance).

Thresholds are different for everyone though.  I might question a coin afterr 50 heads, whereas someone else might question it after 10 heads and someone else only after 10,000 (never).  In magic, someone might question 2-3 unlikely hands while someone else may never question.  I think both extremes are off the mark.

[Harlequin]

Taking a side detour for a second:
If a coin flips up heads 10,000 times in a row, you better believe I'm going to not think the chance of flipping tails on the next flip is 50:50.  The 50:50 figure is based on the assupmtion of a fair coin.  The fair coin is like the "frictionless plane" in physics or the "0 resistance wire" in electronics.  It's decent for modeling, but does not exist in reality.  You can use statistics to find the likelyhood that the coin is fair, and in this case that likelyhood is very low (though still possible).

Its even one step more abstract than your frictionless plan or 0 resistance wire.  If you had a plane or a wire, you could test that plane for the existance of friction or the existance of resistance.  So in our known universe, we have not tested any planes are that frictionless and we have not tested any wires that have no resistance.  However there is no test that will prove or disprove* a hypothesised probability model.  As I said before, believing in probabilty is one thing, but proving it is impossible (much like God).

*- well I guess you could disprove a 0% chance probability model with a single sucess, but outside of that you can't. 

Back to the coin experiment, where you flip 10,000 heads in a row.  It is very likely the coin has a bais towards heads.  But theoretically the coin could actually favor Tails 1,000 to 1.  And the Heads got "lucky" 10,000 times in a row.  So even though we have observed 10,000 heads in a row, it has not proven or disproven a Head's bias (or a tails bias).  Where in the frictionless plan example I can say "well I fell on the plan and scraped my knee, therefor it is not frictionless."

The other mindscrew on this, is that you actually don't use Statistics to derive likelyhood.  You use Probability.  Essentially you theorize a probability model for all probability models; then adjust the model to find the most likely probability model given the statistics you have collected (its call regression analysis). 

Statistics referes only to observations we have gathered about the past.  It is Tangeble and Scientific.  It is not predictive, or determinate.  Its a history of what was rolled up into neat packages (mean, mode, range, etc).

Probability is built only of mathematics, logic, and assumption.  It is the lense we use to predict future events, scrutinize experiance, and distill statistics into something useful.  It provides a simultanious glimpse of everything that could be and could have been.

Luck is as undefinable as Faith.  Its a personal messure we use to comprehend and describe our world.  For example, I have to walk about a block and a half from the parking lot, to my office building in Hartford.  Today I didn't get Mugged.  How would you explain -why- I didn't get Mugged.  Was it a Miracle?  Did Jesus Protect me?  Statistics show it for me (a 20-30 year old male, in broad daylight) Its extremely unlikely to be mugged.  Was I lucky?  Was I just "not unlucky"?  Another statistic shows that for me (a person named Jeff Carpenter on Friday at 9am in Fall) its a has never occured.  Was it Impossible for me to be mugged?

[arctic79]

I can say without a shadow of a doubt that it is not impossible for you to be mugged.  The probablity of you being mugged is low because I don't live in Hartford, the probability of you being mugged would increase 100% if I moved to Hartford.  Lucky (there is that word again) for you the probability remains low for the probability of me moving to Hartford is low.

I agree that luck is as undefinable as faith.  As neither has been proven, yet are accepted at different levels by the individual.  This of course makes it near impossible for either to be proven because of the individual biases that exist and the preexistance of the thoery of probabilty.  Just like a frictionless plane, we have yet to encounter it so how can or would we prove it.

[The Acidic Hasidic]

I am actually very surprised at the level of intelligence in this thread. At this point in time, on the other forums I post on (non magic forums) this thread would have turned into YOU GUYS ARE GAY FOR TALKING ABOUT MATH ON DANCING GAME FORUMS or YOU ARE WRONG I AM RIGHT I HAVE A PHD IN BLAH EVEN THOUGH I AM ONLY 14 AND IN HIGH SCHOOL. So yeah its nice to see some intelligent discussion for once.

But can only attest to mathematical probabilities (its what I do for a living... kinda).

We flip a fair coin because it eliminates chaos from our system. In reality no coin is fair, nor is the person flipping it. But we forget about these variables because it allows us to speculate and predict outcomes. The reason why mathematicians do not like to study the Go game in its entirety is because of its random and chaotic nature. So we wait till the games go on, and moves get predictable before we start making formulas (which never ever hold btw).

 Shuffling. This is the key randomizing element in the game, but it is never truly random.  We assume it is because we can better analyze the game this way. It makes our calculations simpler.

Luck is just another chaotic element of probabilities. In mathematics we call this an anomaly. Or you can see at as part of The Law of Large Numbers. What are the odds of flipping 52 tails in a row on a fair coin? It’s a very small number. But on a large enough scale it is guaranteed to happen.

I guess the point I am trying to make here is that its not great to talk about magic based purely based on probabilities, because they will ALWAYS be wrong. Nobody has the time, patience, or mental capacity to deal with such a large system, as is, the game of magic.  My essay ‘On Card Control in Magic’ (which I am in the process of scanning into my computer... be patient) doesn’t deal with probabilities of individual events that happen in a game of magic the gathering. Its based on an analysis of what card control does in the game, based on  probabilities in different situations (that may be a little confusing but I really don’t know how to explain it any other way).  And that whole essay turned out to be a dud too.

Its ok to discuss probabilities of this game, and in a perfect world where we all have perfectly random shufflers at our disposal and people have no control over card order, these probabilities would be correct. Sometimes it useful to talk about this, but not always. Sorry this post drifts a lot... im not a great writer :<

[arctic79]

I couldn't agree more with your post.  I just couldn't find the right way to say it, and I think you did well with an explaination. 
The acknowledgement of anomalies is important as well as the flaws of shuffling.
Your essay sounds like it might be worth a read.

[Harlequin]

I agree that luck is as undefinable as faith.  As neither has been proven, yet are accepted at different levels by the individual.  This of course makes it near impossible for either to be proven because of the individual biases that exist and the preexistance of the thoery of probabilty.  Just like a frictionless plane, we have yet to encounter it so how can or would we prove it.

However we can abundantly prove that a surface is -not- frictionless.  My point was basically that in probability, no matter how large the sample statstics you can never even disprove a (non-zero) probability model. 

[arctic79]

I agree that luck is as undefinable as faith.  As neither has been proven, yet are accepted at different levels by the individual.  This of course makes it near impossible for either to be proven because of the individual biases that exist and the preexistance of the thoery of probabilty.  Just like a frictionless plane, we have yet to encounter it so how can or would we prove it.

However we can abundantly prove that a surface is -not- frictionless.  My point was basically that in probability, no matter how large the sample statstics you can never even disprove a (non-zero) probability model. 

That depends how far outside the box we can think.  As knowledged as we are with science and the laws of science, one could thoerize that there are elements in the universe that break the basic laws of science.
I understand exactly what you are saying.

Back to Magic, just how serious does anyone take the game that they have to do probability calculations to determine 4x Card "A" or 3x Card "A" outside of an exercise in math.  I know there is one writer on SCG that has beat the topic to death in the driest of articles, and I really wonder how much fun is lost in their magic experience by doing this.  I find it more fun to deck test various compositions and some times even ignoring probability altogether just for the heck of it.  Of course if math is your field of interest then I understand, my father used to think doing math with flash cards was fun quality time...yeesh....some people!

[LordHomerCat]

I agree that luck is as undefinable as faith.  As neither has been proven, yet are accepted at different levels by the individual.  This of course makes it near impossible for either to be proven because of the individual biases that exist and the preexistance of the thoery of probabilty.  Just like a frictionless plane, we have yet to encounter it so how can or would we prove it.

However we can abundantly prove that a surface is -not- frictionless.  My point was basically that in probability, no matter how large the sample statstics you can never even disprove a (non-zero) probability model. 

That depends how far outside the box we can think.  As knowledged as we are with science and the laws of science, one could thoerize that there are elements in the universe that break the basic laws of science.

I don't understand the relevance of what you're saying?  Harlequin is stating the difference between the theoretical frictionless plane vs statistics, and how those are much different assumptions to make.  Because of how statistics work, you can never disprove stuff just as he said.  Therefore, you can never really test a system to see if it is truly ruled by the probabilities, because there's no sample large enough to get a definitive answer.  The frictionless plane (and other theoretical concepts like it) is difference because you can disprove it pretty easily.  All you do is give a counter-example; that's impossible for non-zero probability (regardless of "thinking outside the box").

I'm glad you like to have fun and all that, but to explain how you like to ignore probability and just do whatever you feel like is exactly what the study of statistics and probability will allow us to avoid.  By using a scientific basis, we should (theoretically) be able to come up with 'better' (more easily predictable and playable) decks.  It's fine if you aren't interested in that, and the problem of statistics means that even the perfect deck can just get 'unlucky' and lose a lot, if you hit a statistically unlikely patch, so it's hardly a perfect science, but I see no reason to try and discourage other people from that approach.

[arctic79]

I don't ignore probabilty.  But let's be honest here.  How much intelligence and research is required to understand that 4 of in a deck will increase your odds of getting it in your opening hand then 1,2, or 3 of said card?  To waste hours upon hours crunching numbers that tell you the same thing you already know instinctively is an act of futility. 

In no way am I trying to discourage people from trying to achieve anything.  It just miffs me that some people continue to chase something they already know.  Like I said if they enjoy doing this, that is up to them, if they gain something from it that is great.  But to persue a holy grail type of perfect deck in an imperfect system is borderline lunacy.

[SiegeX]

If a coin flips up heads 10,000 times in a row, you better believe I'm going to not think the chance of flipping tails on the next flip is 50:50.  The 50:50 figure is based on the assupmtion of a fair coin.

This is what I get for not making that assumption up front, fair enough.  But the point I was trying to make was that all cheating aside, the 50:50 probability of the next flip is not causal, i.e there is no sense of being 'due'

However there is no test that will prove or disprove* a hypothesised probability model.  As I said before, believing in probabilty is one thing, but proving it is impossible (much like God).

I think this applies to pretty much all of mathematics as they all start with some axioms in some form or another.  Without having the God view/perspective, all of our theories and models rely in taking some things as granted in order to define others which help explain the physical workings of our universe.

The fair coin is like the "frictionless plane" in physics or the "0 resistance wire" in electronics.

Well, superconductors are for all intents and purposes zero resistance wires.  I don't know for sure if the Meissner effect *requires* the resistance to be zero for it to happen, but I do know that all of our measurements using our latest technology say zero to date.

I am actually very surprised at the level of intelligence in this thread. At this point in time, on the other forums I post on (non magic forums) this thread would have turned into YOU GUYS ARE GAY FOR TALKING ABOUT MATH ON DANCING GAME FORUMS or YOU ARE WRONG I AM RIGHT I HAVE A PHD IN BLAH EVEN THOUGH I AM ONLY 14 AND IN HIGH SCHOOL. So yeah its nice to see some intelligent discussion for once.

I wouldn't say I'm surprised more than I am delighted.  There are some very intelligent people on this forum (as evidenced by this thread) and its what keeps me coming back to this site.  This is quickly becoming one of my favorite threads thus far.

[SiegeX]

I don't ignore probabilty.  But let's be honest here.  How much intelligence and research is required to understand that 4 of in a deck will increase your odds of getting it in your opening hand then 1,2, or 3 of said card?  To waste hours upon hours crunching numbers that tell you the same thing you already know instinctively is an act of futility.

We do this because we only have 60 slots to fill and we want to optimize these slots effectively.  Sure we know inherently that putting 4 copies of a card will give us our greatest chance of seeing it, but maybe 3, 2, or 1 is 'good enough.  By looking at how fast the metagame is, we can determine what our fundamental turn should be such that we still have a decent chance of being alive.  Using this info, we can then vary the card quantities such that the probability of drawing them is at the optimal point of being very likely but not overkill.  This allows us to open up slots for answers/strategies that we would not have access to otherwise.

[Laplie]

@SiegeX, The Acidic Hasidic:

I wasn't saying that we shouldn't use probabilities to determine outcome cause we don't have perfect systems.  I think "fair coin" and "totally random deck" are great ways to simplify and model the situation.  I just wanted to take the thread aside for a sec to say that in real life, we shouldn't be blind to the world.

I don't ignore probabilty.  But let's be honest here.  How much intelligence and research is required to understand that 4 of in a deck will increase your odds of getting it in your opening hand then 1,2, or 3 of said card?  To waste hours upon hours crunching numbers that tell you the same thing you already know instinctively is an act of futility. 

It doesn't take hours, it takes 10 minutes max by calculator, even less if you use a tool like Excel or MWS's deck analysis suite.  And once you know 39.94% for a 4of in opening hand, that number doesn't change.

Knowing probability doesn't just help with deck construction, it also helps while you're playing.  Being able to ballpark certain calculations can help you immensly.  Knowing that the expected value of your next flipped card puts you down too close to 0 life tells you to kill dark confidant now rather than next turn.  Being able to ballpark how many more turns until you combo win tells you whether you have to deal with your opponent's creature threat or not.  Some people get these numbers by just playing their deck over and over till they know it.  The problem with just using previous games is that once you change a card or two, your earlier testing becomes slightly inaccurate.

Poker players and BJ players use probability to great success, magic players should as well.

[John Jones]

Sorry for the up, but you guys also forget how much your deck loves you or does not love you.

I personally can keep no mana hands because I know my deck loves me and I will draw into a mana source.

You also have to have a feel for the cards. Everyone knows that if your unlucky friend comes over to watch your game, the top cards of your opponents deck change (unless you have already seen them) and the top cards of your deck change (unless you have already seen them). Since your friend is unlucky and your deck does not love you, you will not draw well and inevitably lose the game.

Now to counter all of this, you could stack your deck. That is cheating but it could work.

I personally don't have to stack my deck because my cards and deck love me like my mother.
Also, a reason someone Top decks that Will or draws that game winning bomb is because they have the love of their deck.

[Mantis]

@John Jones:
Are you for real? I certainly hope not.

You remind me about one of my playtest buddies who always exclaims how much bad luck he has, while actually he just needs to improve on his game. The best thing that ever happened to me and made me go to the next level as a player was get rid of believing in luck. The goal in a game of Magic is to decrease your chances of losing the game as much as possible, if one can do this flawlessly I will tell you that even the most 'unlucky' person alive will perform above average and probably T8 or even win just about every tournament he enters (given a viable deck ofcourse). I mean are you telling me that Luis-Scott Vargas, Raphael Levy and Kenji Tsumara are just 'lucky'all the time? Oh and why is it that the technical masters who spend countless hours on tuning their decks and perfecting their play always appear on the higher tables?, and the bad players never make T8 and always attribute all of their losses to luck?

Because:

SKILL > LUCK.

Also, please keep no mana hands if we ever get to play against each other and I will show you that my deck loves me more.

EDIT: Mods, if my post is derailing this thread, by all means feel free to delete it.

[Korhil]

This moves away from the original thread topic, but it seems it's moved a bit anyhow.
(btw, interesting stats from the people that have posted data.)

With regard to Magic as a game, I think it is important to pay some attention to probabilities and have a fair understanding of what a 'correct' number of lands to run are or how many duress effects to play.
I don't think any deck has been perfect from its first build however. You may have applied all the best theory available in assisting you with the mix of cards in your deck, but when you start testing you find that it's not as 'perfect' as it should be.

Things that can make it seem less perfect in testing is that your opponent is making assumptions about the distribution of your deck and therefore the resources you have available. They are assuming that you’re playing a specific configuration of spells and they will play their spells in a way to disrupt your game.
There isn't always a correct order to play things in because of the interactive nature of the game. If you make what probability suggests is the best action, there’s a reasonable chance your opponent has expected that line of play, and will have done their best to be in a position to disrupt it when your commit to it.

What I'm trying to suggest, is I believe intuition is important.
You want you opponent to believe things about your deck, and play to avoid them.

Concrete examples are difficult to come up with.

One that begins to show the concept I have is:
Your opponent believes you are playing with Daze in your deck as a 4 of when you aren't, in fact you may have sideboarded them all out. They end up playing much of the game trying to play around the spell. They think they are playing correctly and playing their spells in a way that is 'probably' going to win them the game. You assume they know you would sideboard daze out for this match up, and don't know to take advantage of the situation and bluff having daze more often.
Both of you are playing thinking you have perfect information of the game and are making 'correct' decisions.

Equally, you may be playing with 4 daze in your deck when you should have sideboarded them out. Your opponent doesn't think you have any in your deck at all. Your opponent plays with what you consider reckless abandon, throwing bombs at you while tapping out. You never have daze in hand to punish him and get destroyed.
You'll likely think he played badly and you were unlucky to lose.
They will think that they won as they should have because you didn't bring in tormods crypts for your dazes. (No good example of what would be better comes to mind right now, assume there is a match up where some card is definitely better than your dazes would be) 

Magic isn't a game of perfect information like Chess.
You may be playing a game making decisions for the wrong reasons. I think there is a limit to how much you can apply probability to Magic. Intuition and feel for the game can be more powerful tools.

---Korhil

[Diakonov]

I know that this thread has long been off the original topic, but I created a quick formula to calculate odds of drawing at least one of a card you want out of any number of cards drawn, for those who are unfamiliar with any pre-programmed computer-based functions:

x = number of copies of card you want to see in the deck (i.e. 8 duress effects, etc.)
y = number of cards drawn

1 - (60-x)!(60-y)! / 60!(60-x-y)! = probability

So, for example, likelihood of drawing at least one Bazaar of Baghdad (4 of) in the first 7:

1 - 56!53! / 60!49!

doing some canceling...

1 - 53*52*51*50 / 60*59*58*57

1 - .6005

.3995 = 40%

Throw any numbers you want in for x and y and repeat.  Nice little pen & paper method.