Pioneering a Manabase Using Probability and Statistics

One of the most difficult things to master in this game is the art of deck building. It is, in my opinion, the most challenging element to the puzzle that is Magic. You can find plenty of content addressing the following questions: “What is the expected metagame?”, “Where do I find my upside?”, “What is my metagame edge?”, and “How do I ‘break’ it?”

These are all very important questions to consider but don’t address the most important concept related to deckbuilding; the manabase.

When it comes to deckbuilding, I consider the manabase to be foundational.  With a house, a weak foundation will ruin all the beauty, planning, and effort that you have put into your designs. When it comes to a manabase, we need to build a strong foundation so that we can make sure we are going to hit our colors on time to cast our spells. We also may want to select lands that provide some extra utility so that we can player higher land counts and mitigate the effects of flooding out.  Obviously, the more lands we include, the fewer chances we will have to feel mana deprived during the game but too many lands might result in some inaction on the other end of the spectrum.

Today, we are going to go into the theory behind how manabase math works, as well as discussions about how to hit some very critical flashpoints for manabases to support spells in your decks as you explore the vastness of the Pioneer format.

Theorycrafting of Manabase Mathematics

Rrrrrrrrt…let me pull this thing over for a second.  If math freaks you out, you may want to skip right to the next section where I talk about practical application.  If you’re genuinely interested in finding out how my findings are calculated.  Stay put.  Ok, seatbelts back on…

To calculate manabase mathematics, we’ll need to apply probability theory and statistics. We are investigating the probability (likelihood) of s successes (this probability theory measures things in a binary fashion, they can either be successes or failures) in n random draws without replacement (meaning you draw cards and they don’t get shuffled back into a deck) from a finite population N that contains exactly S objects that have a condition that can be considered a success (for instance, the total number of lands included in the deck). In mathematics, this is defined as a hypergeometric distribution.

A good example of this (which will be a bulk of the investigation in our practice section) is that you would like to draw 2 lands (s = 2) in your opening hand from a 24 land deck (N = 60, S = 24, n = 7). Our “success condition” is considered to be that the card is type “land” and we have stated that our goal is to draw 2 of them.  The probability of an exact condition like this happening can be calculated in the following way:


Where:

  • N is the population size
  • S is the number of success states in the population
  • n is the number of draws (i.e. quantity drawn in each trial)
  • s is the number of observed success
  • {a \choose b} is the binomial coefficient

This equation, when everything is plugged in properly, will provide you a decimal that is less than 1.  In the probability theory of mathematics, the summation of all possible probability events in a given universe is 1. This makes reasonable sense because of the concept that if you take every possible combination of drawing a number of a type of card (0 to 7), you will cover 100% of the possible hands that you might draw in a game. Mathematically this identity is written as this:


For those not well versed in the mathematical fields, the sigma (Σ) means to add together all of the possible answers to that equation based on changing the number of successes (s) from 0 to the total number of draws.

The goal of all of this is to identify, specifically, what we tend to look at when we build decks, which is a desire to draw a balanced mix of lands and spells.

What happens when we want to be more specific, though? What if I want to draw a certain number of cards of two different types, such as untapped lands and one-drops? By doing this, we begin to examine what mathematicians describe as a Multivariate Hypergeometric Distribution. This is calculated by viewing the different number of success conditions and numbering them using the generic numbering of i =1 (the first kind of card) to i=c (the total number of card types). Then total population involved here (N) is calculated as:


Where:

  • S  is the total number of success for each condition
  • i is the number classification of the success condition
  • c is the total number of success condition classifications

The probability calculated from these variables is calculated as:


Where:

  • N is the population size,
  • Si is the number of success states in the population of the classified success type
  • n is the number of draws (i.e. quantity drawn in each trial),
  • si is the number of observed successes of the observed success type.
  • {a \choose b} is the binomial coefficient
  • i is the number classification of the success condition
  • c is the total number of success condition classifications

Similar to Sigma, Capital Pi (Π) is the product (multiplication) of terms over the numbers on the bottom and top of the symbol.

We have covered all of the mathematical concepts connected to this concept we are going to put to use! Now let’s get down to business and do the calculations!

Applications of Manabase Mathematics

Okay! Glad that everyone has made it through the theory alive. Now let’s put the theory into practice! There are two online tools I utilize to obtain all of the numbers provided in this article. For reference, here they are:


Manabases

Let’s start with the plain and simple manabase math concept. There are a wide array of lands that provide various colors, come into play tapped and untapped, and fuel your gameplan. The most important thing to do is to balance your ability to produce colored mana based on what you need to be able to cast throughout the game. The simplest way to do this is to use total the mana symbols on the cards in the deck to get an idea for the number of mana sources one might need in a deck. This is done simply. Take the following deck for example:


Hypothetical Mono Black

Creatrues (40)
20 Bloodsoaked Champions
20 Kytheon, Hero of Akros
Lands (20)
20 Lands

In this deck you would count up and say that you have 20 black symbols and 20 white symbols and would need an even split of lands. This is a simple theory, but there is more to be said about this.

This deck operates with trying to play a 1 drop of either color so you would like as many lands that make both colors as possible. Even though the symbols are split right down the middle, it might incentivize you to play dual lands to hit either of your one-drops. Thus, you might consider 4 Godless Shrine, 4 Concealed Courtyard, 4 Caves of Koilos as the base of your deck since they make either color you might need.  Mana Confluence, too, does what we are looking for.  We could also include 4 Unclaimed Territory because both of our creatures are humans so it could provide mana for either one on the first turn. Based on the goal of having access to the right mana sources to cast either creature with an untapped land on turn one, this would be considered a perfect manabase for this deck.

Then, you can take your hypergeometric distribution and ask the questions, “Is 20 the correct number for lands vs. spells?”, and “Am I willing to sacrifice some mana consistency to play cards like Mutavault or Castle Locthwain to improve my late game?”

You can see that when you start to consider these options, things start to spin out of control and become amazingly complex. We are going to cover the basics of color sources to cast/enable cards below, but when it comes to the higher complexity items like using Castle Locthwain vs. Swamp number X, that is something that the deck builder has to decide based on past experience with the game and metagame preference.

Again, this math is for setting the foundation and to make sure that the things you put into practice will work as intended. The specifics of how to utilize this mathematics along with our knowledge of the card pool will come in a follow up article, but for now, let’s focus on the important keystones of the format, and how we can shape our deckbuilding to make them a possibility.


Turn One Thoughtseize

There are a lot of questions that come to mind when we discuss the impact of turn one Thoughtsieze. A lot of players see this as a premier play for their midrange, and sometimes aggressive, strategies. When building one’s manabase, you have to decide what you would consider an acceptable fail rate for hitting Thoughtseize on turn one and understand that not every game will start like that. Let’s take a sample mono-black aggro deck, and break down from there conceptually what the mana math looks like.

One of the most recent mono-black PTQ winners on MTGO has a list that looks like the following:


Mono Black Aggro [Pre-Smuggler’s Copter Ban]

Creatures (24)
Bloodsoaked Champion
Knight of the Ebon Legion
Murderous Rider
Night Market Lookout
Rankle, Master of Pranks
Scrapheap Scrounger
Spawn of Mayhem

Spells (8)
Thoughtseize
Fatal Push

Artifacts (4)
Smuggler’s Copter
Lands (24)
15 Swamp
Urborg, Tomb of Yawgmoth
Castle Locthwain
Mutavault

Sideboard (15)
Dark Betrayal
Duress
Kalitas, Traitor of Ghet
Noxious Grasp


In a deck like this, we can expect to be able to draw Thoughtsieze on turn one on the play about 40% of the time (39.949%). This includes the potential to draw more than one copy of the card.

If we use a single variable hypergeometric distribution and remove a card from our opener (assuming a six card opening hand with a 59 card deck — as if we set aside a Thoughtsieze included in our opener), we would want to consider the rest of the hand, specifically, whether or not we have an untapped black source we would need to cast this card (or in this aggressive deck, one of our other one-drops) on turn one.

With the defined variables above, we have N = 59, s =+1, S = number of potential successes (we are going to make this variable to find percentage ranges) and n = 6 on the play or 7 on the draw, we will get the following percentages in tables:

On the Play

Number of Untapped Black Sources Percentage Chance of Success
13 79.211
14 81.923
15 84.333
16 86.47
17 88.358
18 90.021
19 91.481
20 92.759

On the Draw

Number of Untapped Black Sources Percentage Chance of Success
13 84.31
14 86.698
15 88.767
16 90.554
17 92.092
18 93.41
19 94.535
20 95.491

I listed the row with 16 black sources in bold because it represents the amount of black sources present in the mono black deck shared above. Based on your risk tolerance, you can tweak the percentages for how many untapped sources of black you have to make this happen.

However, in a format like Pioneer, where we don’t have fetch lands and are left to alternatives like Fabled Passage, you won’t be able to include as few colored mana-producing lands to leave space for more utility lands after playing fetches. To enable 20 turn-one black sources in a mono color deck, it is very likely that you will need to play 20 Swamps, which then takes away from your long game.  This is a decision that every player will need to make. This particular pilot decided to sacrifice 8-10% consistency in casting their one-drop on turn one to be able to have mana sinks in the mid- to late-game. Whether or not that is correct is left to the risk tolerance of the deck builder.

If we expand this to the entire deck and say that we want to keep our current manabase (16 untapped black sources total) and want to calculate the percentage of time we have a one-drop on the play or draw, we could use a multivariate hypergeometric distribution to figure out at what point do we hit the correct combination of one-drops and untapped sources to meet our risk tolerance:

On the Play

Untapped Black Sources One Drops (proactive) Percentage Chance of Success
15 14 75.1
15 15 77
15 16 78.7
16 14 76.7
16 15 78.7
16 16 80.5
17 14 78.2
17 15 80.2
17 16 82

On the Draw

Untapped Black Sources One Drops (proactive) Percentage Chance of Success
15 14 81.7
15 15 83.4
15 16 84.8
16 14 83.1
16 15 84.8
16 16 86.3
17 14 84.3
17 15 86
17 16 87.5

Here we see that the PTQ-winning deck had a proactive one-drop (11 creatures and 4 Thoughtsieze) 78.7 percent of the time on the play and 84.8 percent of the time on the draw.  If that player always thought that they needed to get out of the blocks doing something proactive on turn one, then they would have a relatively high fail rate (1 out of every 4-5 games) but it is more than reasonable to accept this fail rate depending on your hand texture or match-up. The math here gives you the theory behind how often you will succeed, but it is always up to your risk tolerance and experience of the game to put the theory to good use.


Turn Four Supreme Verdict

This one is where we are going to flex the power of multivariable hypergeometric distributions to the maximum. Often times, our UW control decks in Pioneer are dependent on Supreme Verdict as a catch-up condition after potentially setting up some early card advantage with something like Search for Azcanta, or Narset, Parter of Veils, or Teferi, Time Raveler and will be looking to clear the board to pave the way for Teferi, Hero of Dominaria or the ability to cast Granted to find a powerful win condition out of their sideboards, and this card is very pivotal to the success of these decks. You need to pack interaction along the way in order to survive long enough to cast your catch-up mechanism.  This can be very difficult. You will obviously play removal, but let’s just examine how likely you are to achieve the feat of casting Supreme Verdict on time. This becomes difficult to calculate because we often play lands that can provide both blue AND white, instead of producing just one of the two colors.

Consider the following UW Control Deck from an MTGO League


UW Control [11/2019]

Planeswalkers (10)
Elspeth, Sun’s Champion
Jace, Architect of Thought
Narset, Parter of Veils
Teferi, Hero of Dominaria
Teferi, Time Raveler

Spells (22)
Supreme Verdict
Absorb
Azorius Charm
Blessed Alliance
Dig Through Time
Opt
Sphinx’s Revelation
Syncopate

Enchantments (4)
Cast Out
Seal Away
Search for Azcanta
Lands (24)
Castle Ardenvale
Castle Vantress
Fabled Passage
Field of Ruin
Glacial Fortress
Hallowed Fountain
Island
Plains
Temple of Enlightenment

Sideboard (15)
Aether Gust
Detention Sphere
Dovin’s Veto
Lyra Dawnbringer
Mystical Dispute
Nyx-Fleece Ram
Pithing Needle
Rest in Peace


In this deck, we have quite a few dual lands, so there are going to be some amount of complication. I am going to assume that we perfectly sequence our land drops so that our turn four land comes into play untapped to allow Supreme Verdict. With that in mind, lets take a look at the numbers on the deck above, as well as a couple of different compositions, so that you can get a view on how likely this is to happen. The issue is that you are limited by the number of sweepers that you are playing, which I will keep constant at four.

There is some mathematical discussion that needs to be had about how this gets calculated. So, first, we would need to calculate all of the following probabilities that represent each combination of four lands that would allow for 1UWW:

  • 4 duals
  • 3 duals and 1 solo white
  • 3 duals and 1 solo blue
  • 3 duals 1 colorless
  • 2 duals and 2 white
  • 2 duals, 1 white, and 1 blue
  • 2 duals 1 white 1 colorless
  • 2 duals, 1 blue, and 1 colorless
  • 1 dual, 1 white, 1 colorless, and 1 blue
  • 1 dual, 1 white, 2 blue
  • 1 dual, 2 white, and 1 blue
  • 1 dual, 2 white, and 1 colorless
  • 2 white, 2 blue
  • 2 white, 1 blue, and 1 colorless
  • 3 white, 1 blue

Then we would need to subtract the potential overlaps in these calculations.

On the Play

Number of Duals Number of Blue Number of White Number of Colorless Number of Supreme Verdict Percentage Chance
13 7 3 1 4 28.2
12 7 4 1 4 28.1
12 7 3 2 4 27.2

On the Draw

Number of Duals Number of Blue Number of White Number of Colorless Number of Supreme Verdict Percentage Chance
13 7 3 1 4 36.3
12 7 4 1 4 36.2
12 7 3 2 4 35.2

As you can see, the odds of hitting Verdict on time aren’t the highest, so you need to have functional side interactions to be able to help control the rest of the game.  I would assume that, because of how important that the need of having Supreme Verdict is for the control decks, I don’t think we will see a strictly UW deck find prominence without a relevant and mana-efficient spot removal spell similar to Fatal Push.


Untapped Check Lands/Reveal Lands Viability

For reference, I consider the “check land” cycle to include cards like Glacial Fortress and “reveal lands” to include cards like Port Town.

This one is going to be a little bit less intense, and will appear similar to the Thoughtsieze section insofar as we can approach this from two different directions. The first is to take out the check/reveal land and assume that it starts in our hand.  Then, if we consider the need for enablers (i.e. basic Plains/Island for Glacial Fortress), we must require a hand with exactly check/reveal land plus an enabler.

Consider the following deck from the SCG Invitational:


Simic Nexus [11/2019]

Spells (25)
Censor
Cyclonic Rift
Dig Through Time
Fog
Growth Spiral
Nexus of Fate
Opt
Supreme Will

Planeswalkers (4)
Narset, Parter of Veils
Tamiyo, Collector of Tales

Enchantments (6)
Search for Azcanta
Wilderness Reclamation

Creatures (1)
Thing in the Ice
Lands (24)
Blast Zone
Breeding Pool
Castle Vantress
Fabled Passage
Forest
Hinterland Harbor
Island
Lumbering Falls

Sideboard (15)
Cerulean Drake
Lazotep Plating
Mystical Dispute
Narset’s Reversal
Negate
Return to Nature
Spell Pierce
Swan Song
Thing in the Ice
Tireless Tracker


First, lets calculate the chance that we have at least one land in our opener to enable a check/reveal land to come in untapped:

On the Play

Number of Enablers
Percentage Chance of Success
11 72.765
12 76.169
13 79.211
14 81.923
15 84.333
16 86.47
17 88.358

On the Draw

Number of Enablers
Percentage Chance of Success
11 78.417
12 81.565
13 84.31
14 86.698
15 88.767
16 90.554
17 92.092

Next, here are the odds to have both pieces (check/reveal land PLUS an enabler):

On the Play

Number of Enablers Reveal/Check lands
Percentage Chance of Success
11 4 29.5
12 4 30.8
13 4 32
14 4 33.1
15 4 34
16 4 34.9

On the Draw

Number of Enablers Reveal/Check lands
Percentage Chance of Success
11 4 35.3
12 4 36.7
13 4 37.9
14 4 38.9
15 4 39.8
16 4 40.6

Based on this math above, check lands and reveal lands will often end up coming into play tapped in this format a higher percentage of the time than we would like from turn one.

If you can afford to include some temples in your deck to improve your draws (ie. Temple of Enlightenment), it might be worthwhile, but it comes again to the balancing of how often you need your lands to come into play untapped, fix your mana, and when.  Obviously, a card like Temple of Enlightenment would come into play tapped 100% of the time, whereas, later in the game, something like Glacial Fortress becomes increasingly more likely to enter untapped.


Fabled Passage

Fabled Passage is Pioneer’s most effective fetch land.  For this one, we want to confirm that we are sequencing such that we can play it on turn four.  Because of this unique condition, Fabled Passage necessitates a very specific calculation. The calculation for this involves how often it appears by turn four, and when it does, how often will you end up fetching an untapped land when it matters. The table below will tell you, based on your Passage count, how often you will draw a Passage based on the number of Passages in your deck.

On the Play

Number of Passages Percent chance to draw Passage by turn four
1 16.667
2 30.791
3 42.724
4 52.772

On the Draw

Number of Passages Percent chance to draw Passage by turn four
1 18.333
2 33.559
3 46.16
4 56.55

Next, based on number of Passages and lands, how often will you enable a turn four Passage on the play and draw:

On the Play

Number of Passages Land Count
Percent chance to have an active Passage on turn four
1 21 10.9
1 22 11.6
1 23 12.2
1 24 12.8
1 25 13.4
2 21 18.9
2 22 20.3
2 23 21.6
2 24 22.8
2 25 23.9
3 21 24.6
3 22 26.6
3 23 28.5
3 24 30.2
3 25 31.9
4 21 28.1
4 22 30.7
4 23 33.1
4 24 35.5
4 25 37.7

On the Draw

Number of Passages Land Count
Percent chance to have an active Passage on turn four
1 21 13.5
1 22 14.2
1 23 14.8
1 24 15.4
1 25 15.9
2 21 23.5
2 22 24.9
2 23 26.2
2 24 27.3
2 25 28.4
3 21 30.6
3 22 32.6
3 23 34.6
3 24 36.3
3 25 37.8
4 21 35
4 22 37.8
4 23 40.3
4 24 42.6
4 25 44.8

With this in mind, it seems that there is a higher percentage of games that could play out where you don’t draw Passage vs. games where you do draw Passage but not enough lands, as the number of Passages you can include is obviously limited to four copies. Based on this information, though, we can see the percentage of games where you draw a Passage and the percentage of games where you have an active passage on four when you need it.  You could use some other probability theory work to reverse engineer how often the Passage will fail you and have to fetch a tapped land, instead of choosing to sequence it ideally.


Closing Remarks

I appreciate you reading through this!  Stay tuned for a part two to this series in which I will look at the real estate that pioneer has to offer and talk about how to use the simple manabase math along with some of the check land, Fabled Passage, and early play math we have considered to try and shape some decks with positive early game strategies.  I’ll also share some examples of how you can build your deck and streamline your manabase to fit your spells.

Leave comments below or reach out by emailing us at cardknocklifepodcast@gmail.com.

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