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By: PHahn, Peyton Hahn
Apr 09 2014 1:05pm
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Mulligan: Art or Science?
 
Peyton Hahn
 
              So there I am: submitting my sideboard changes and entering Game 3 of the 2nd Round of an 8-man Standard tournament. I’m on the draw, playing with GB Dredge against Mono-Black Devotion, and here’s what I see:
 
 
Forest Elvish Mystic Herald of Torment 
Nemesis of Mortals Grisly Salvage Commune with the Gods Sylvan Caryatid
 
             Do you keep?  This hand has the potential to have an amazing start but also only has one land.  However, we do have Elvish Mystic and Sylvan Caryatid to produce mana early if we miss a land drop or two. If my opponent plays Turn 1 Thoughtseize, we might just be dead.
 
             As a player, I’ve already noticed that I am probably looser with my keeps than most.  I kept this hand, reasoning that I could go off if my Mystic landed, and was rewarded with a fairly easy victory.  But this article isn’t about how to beat Mono-Black.  As I’ve begun to play more and more with GB Dredge, I’ve noticed many more land-light (or absent) hands.  So, I wanted to find out: how can I statistically explain the probability of drawing 2 or more land?
 
         Being a stat geek, I am obviously aware that there are formulas to calculate these types of numbers. However, I also understand the “Law of Large Numbers”, which states that over a large sample size, the average of the observations in an experiment will reflect the true expected value. Conditional probability can be tedious and ugly to calculate: the calculated probability of drawing a land in Card 2 of a seven card hand changes depending on whether or not I drew a land for Card 1 (and Card 3 depends on Cards 1 and 2, etc. etc.). If I have an excuse to run an experiment, I’m pretty much taking it anyways.
 
         “But Peyton, there’s no way you can prove anything unless you have a large sample size! You just said so!”
 
         How about 100,000?
 
         My IS major/analytics minor luckily has given me some savvy in setting up an electronic experiment. Thanks to a little Excel skill, a programmer mentality, and some review from a professor, I was able to build a spreadsheet that would simulate the process of drawing a 7 card starting hand, with the value of each drawn card (Land or Non-Land) affecting the probabilities of each subsequently drawn card.  I ran the simulation 100,000 times and recorded how many lands and non-lands were drawn.
 
 
 
 
 
 
              To answer the original question, it looks like I will draw at least 2 lands 75.16% of the time.  That seems pretty high, but if you consider that about 1 in 4 matches I’m going to be looking at an opening 7 with 1 or 0 land, it is a bit more daunting. It is nice to see that the probability of having a starting hand containing 5 or more lands is less than 5%.  Looking at 5 Forests and two Sylvan Caryatids is not where you want to be.
 
              Summary stats for number of lands in the opening hand of a 60 card deck containing 20 lands:
 
 
Mean:  2.32958
Standard Error:  0.0037405
Median: 2
Mode: 2
Standard Deviation: 1.1828571
Sample Variance: 1.399151
 
 
When staring down a land-light opening hand, there are 3 things to consider when choosing to mulligan.
 
1: What are my chances of drawing a land?
 2: How much better is my average mulligan?
                                                                               3: Can I win with this hand?
 
Obviously, “Can I win” should be the most important question, but I can’t measure that with stats. That is the catch-all for other determining factors such as matchup, specific cards in hand, etc.  The first two we can measure.
 
What are my chances of drawing a land?
 
               This is very easy to calculate and can be done with pen and paper.  Start with the number of lands in your deck (in my case 20), subtract the number of lands in hand, and divide by the number of cards remaining in your deck (53 for an Opening 7).  So if I have 2 lands in my opener, my chance of drawing a land during my next draw step is (20-2)/(60-7) = 18/53 ~ 34%.  If I MUST hit my land three drop and I’m on the play, the chances of drawing land in the next 2 turns is  (1 – (.66*.65)) ~ 57%.  
 
            The calculation of LandInDeck/CardsInDeck is also known as land density. This is good to keep track of throughout the game, as you should always play to probabilities. To give a very general example of this, if a situation were to arise where the outcome of a critical decision depends on drawing a land (say, to (Rakdos’ Return) for exactly lethal), it would be good to know the probability that you will draw a land.
 
How much better is my average mulligan?
 
Back to the experiment. Same set up, but one less card drawn per simulation.
 
 
 
 
 
             Well, that’s actually pretty relieving. 66% of the time you mulligan with a 20 card deck, you’re going to start with at least 2 lands.  Unfortunately, you will also draw 0 lands more than 7% of the time.  This is immensely useful when staring down a borderline hand.  
 
Can I win?
 
          I have no statistics to answer this for you (dangit!). However, by knowing your land density and understanding the mulligan probabilities, you should have more information to make a decision that plays to your odds.  
 
          After reviewing these statistics, I think I made the correct decision to keep the one-lander. Not only did I have the Mystic, but I also had a better than 50% chance to hit my second land drop on time if my opponent were to seize my thoughts.  Not to mention the chances of drawing another Mystic.  But, as I said before, I keep a bit looser.  I’d at least say it is defensible.  What do you think?
 
                                                 “Peyton, I don’t play 20 lands, this article is a waste of my time!”
          Okay, you got me. You read this far about land scarcity which probably isn’t as relevant to midrange and control players as it is for aggressive players.  Just to make you all happy (because I love my readers and hope they all win Pro Tours and then give me a shoutout someday), I reran the experiment for 24 land (midrange) and 27 land (control).  
 
24 Land
 
 
 
          Opening Hand Probabilities
 
 
 
 
          Lucky ducks, you’re always starting with at least 2 land! You will face more draws with 5+ lands, but over 70% of your opening hands will have between 2 and 4 cards, which I think is pretty optimal.
 
         Mulligan Probabilities
 
 
           Midrange mulls pretty nicely from a land density perspective.  Half as likely to completely whiff compared to 20 land decks and over 70% of 6 card hands contain between 2 and 4 lands.  I’d feel pretty good about that.
 
27 Land
 
                Opening Hand Probabilities
 
 
          Interestingly, this histogram highly resembles a bell curve, with a center at 3.  27 land players are starting at 4+ lands over 35% of the time, but they suffer much less from the extremes: 1 or less about 9% of the time and 6 or more less than 3%.  Intuitively, control decks will have a much higher land density, which can be a problem when answers/threats are needed. This is why card draw is such an important effect for these strategies.
 
         Mulligan Probabilities
 
 
      What this means for a control player: I’m not sure. I never play control and I don’t really plan on it anytime soon. Rarely seeing no-land mulligans seems pretty good though.
 
Closing Thoughts
 
         Wow, that was a ton of numbers. Even if you don’t play these exact number of lands, hopefully you found this article to be an interesting study of chance within the game.  It’s also important to think about these probabilities when sitting across from any of these archetypes in Games 2 and 3 (once you’ve identified them).  Being able to put your opponent on a certain number of lands in hand can be a powerful tool. I believe that there is still so much to learn about using statistics to improve the quality of play. I hope to continue to find studies like this which can bring some interesting, unique information to my readers and the Magic community at large.
 
        Next week, I’ll be back to update the “Managing Money” numbers: it sure looks like Esper may be losing momentum, as it failed to make Top 8 at GP Phoenix.  We’ll see if this is reflected in the online metagame over the next week. The following week, I’m thinking maybe I’ll do a similar study like this one for the Limited format: 16, 17, and 18 land decks. 
 
        Please leave your thoughts/opinions/criticism below. I take my articles very seriously and want to do everything I can to bring my readers high quality material.  Let me know if there’s any similar studies you would like to see. I am always open to answering any questions regarding how I do the simulations and things like that, but please understand it is difficult to explain.  
 
        As always, thank you for reading. Add me on MTGO and share with anyone/everyone. 
 
 
Peyton
MTGO: shaqdaman

 

 

 

8 Comments

In spite of all your by Paul Leicht at Wed, 04/09/2014 - 14:04
Paul Leicht's picture
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In spite of all your laborious math and excellent reasoning, I am going to go with "it's an artform." I think you make a case for how to reason out hands where you aren't sure but your gut is probably almost never wrong if you play enough (win/lose or draw).

Also intuitively, that starting hand seems terrible if your opponent has the right hand. (If they are on the play, they may have removal waiting for your elf, which isn't necessarily a bad plan if they are able to capitalize on the delay that causes.) Knowledge of your opponent, their style and what they are playing is equally as important as knowing whether you are likely to draw the land you need in a timely fashion. All this not to demean what you have written as it has strong value but for many players I think experience is the key factor in most mulligan decisions, not merely reasoned math. If anything the math should confirm what you gut already knows.

Agreed by PHahn at Wed, 04/09/2014 - 14:31
PHahn's picture

First of all, thanks for reading. I agree that a decision to mull has more to do with math--thus the "can I win" category. What you discussed falls into that. If I knew my opponent was playing burn or rdw, for instance, I'd never keep that hand. Thanks for the input!

Good job once again. This is by IYankemDDS at Thu, 04/10/2014 - 10:05
IYankemDDS's picture

Good job once again. This is an area of my game where I am really trying to improve. I think it takes a lot of discipline to get good at this, because you always hope that those shaky hands are going to get the help that they need. And then sometimes you worry that if you ship a close hand, you'll mull yourself into oblivion and give away the game. I think it is an art! My latest project is to get better at sending back 5-land hands.

Yeah, this article actually by PHahn at Thu, 04/10/2014 - 12:58
PHahn's picture

Yeah, this article actually came out of my own effort to improve my own game in this way. Since writing it, I think my decision making is more sound. Yes, I think it's an art, but I think understanding your probabilities can give you the maximum amount of information to make a good decision.

Bitter old man by Lagrange at Thu, 04/10/2014 - 13:52
Lagrange's picture

People seem to like your article. But allow me to be the bitter old man:

I am struggling to see what new knowledge you are bringing to the table. You should read the first post in the forum thread 'The shuffler thread for a new generation' http://community.wizards.com/forum/magic-online-general/threads/1025716

There Bubba gives you all the probabilities in the bottom tables.

Somehow I also doubt that you are a stat geek. Or at least the simulations are not really impressing me, when the exact formula is easy to use (and implemented in Excel, see HYPGEOM.DIST). 500,000 simulations as stated in the teaser text, when you can just plug 5 numbers into an Excel-function. Had you used and explained the Excel-function then readers could also do their own calculations for other combinations of deck size and land numbers. That would be closer to bringing something new to the table.

Hi Bitter Old Man! by PHahn at Thu, 04/10/2014 - 14:39
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First of all, thank you for reading. Unfortunately, I acknowledge that I have not read every piece of every Magic literature on the internet, so I had not seen the post to which you are alluding. What I do know is that I have not seen anything like what I have presented on this website, or the websites that I frequent. I do not claim to be the only person to have figured out these probabilities (like I said, there are equations and I am sure other people have presented them).

If I wrote this article explaining how to run these simulations in Excel, I don't really know if that adds any value. I'm going on the assumptions that A. People who want this info don't want to do the work to derive it. B. Not all of my readers are Excel savvy. C. Readers understand visualization better than hard numbers.

If A or B are true, then explaining how to create this simulation would be useless. Nobody would care, they just want the results, which is what I'm presenting. Running the simulation through a Monte Carlo simulation with the proper setup and mapping is incredibly difficult to explain to a reader who may have no Excel knowledge. Yes, there are Excel functions that can calculate these numbers, but running a full simulation is not the same thing.

I appreciate you reading the article, and thank you for your feedback, but I disagree with your points of criticism.

First of all, thanks for the by Lagrange at Thu, 04/10/2014 - 18:27
Lagrange's picture

First of all, thanks for the response.

Second, the shuffler thread can be a lot of fun to read and you should frequent the official forums more often, if you had not seen it before.

Third, I am not suggesting that you should explain your simulation. The simulation is over-kill. It is crossing the bridge to get water etc. You get my point. You do simulations, when you cannot easily calculate probabilities from a closed form solution, this is very very very far from the case here. You make it sound like the hypergeometric distribution is rocket science. It is not. It's stat 101. And you could easily explain how to calculate probabilities from that distribution using Excel, so that even non-stat-geeks with no prior knowledge about Excel could do it.

Fourth, don't under-estimate your readers' abilities to read numbers or their interest in how you get to your results. Talking down to your readers, should be avoided. Reading comments from your earlier articles on something about a price-metric(?), people also wanted to know exactly what you did.

Fifth, I appreciate you taking time to explain your reasoning, but I disagree on point A and C. Whereas point B is probably true.

I understand where you're by PHahn at Thu, 04/10/2014 - 20:14
PHahn's picture

I understand where you're coming from. As far as reader intelligence, I'm not trying to talk down or say anyone is stupid. It is just my belief that assuming a less extensive knowledge makes the articles easier to read/understand/etc for everyone. I have taken note of the responses of interest in how I calculate my metrics, and I will definitely be trying to make a point to explaining that. I really do appreciate the feedback/criticism, and I will definitely keep it in mind going forward. I just took some exception to your "doubting" of my statistic interest or ability. Once again, thank you for the feedback, and please continue to give me suggestions in the future.