(Tatsunari, Toad Rider | Art by Justinecruz)

Enchantress and the Frog

Feel like your deck just is not coming together? Welcome to Solve the Equation, where we take a look at the numbers and see what's responsible for your deck falling flat.

Today, let’s build around the new Kamigawa: Neon Dynasty commander, Tatsunari, Toad Rider. We often hear benchmarks of how many cards of a given card type we need in our decks (e.g. 10 card draw and 10 ramp). The inaugural article in this series will take a look at how we can leverage probability to determine how many enchantments we need to reliably advance our strategy. Every deck is different, and benchmarks are helpful, but we can dig deeper and figure out the ratios that really make our deck tick.

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Probability to the Rescue

A factorial of an integer “x!” is the product of all integers less than or equal to x

• n represents the total number of possibilities
• h represents the sample size

We can use this formula to determine how many possible combinations of opening hands our deck can produce. A Commander deck (without Partners) has a deck size (n) of 99 and an opening hand size of 7 (h).

If we plug these numbers into our equation, we find the terrifying statistic that there are 14,887,031,544 possible combinations of the first 7 cards we draw. Our goal is to use math to increase the consistency of a deck with that much variance.

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How Does This Help Me?

First and foremost, it's important to determine your win conditions. Ask yourself, “How do I plan to close out this game?” Commanders such as Tatsunari are fairly open-ended, so it's often helpful to have a primary strategy and a secondary strategy that overlaps with your primary win conditions, as these strategies will inform your deckbuilding. For this build, my strategies are:

• Primary: Play efficient enchantments to drain my opponents’ life with Keimi’s ability before finishing them off with a buffed unblockable creature.
• Secondary: Pull off a combo using the various life gain payoffs that work well with Keimi’s ability

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Using Combinations to Determine Probability (nCr)

Let’s take a look at ramp. Sultai gives us great options for ramp, so we do not have to stretch and we can select the best. I want to reliably have a ramp spell in my opening hand, as getting ahead on mana is crucial, so I want to be at about that 60% threshold at least. However, if we go above that, we increase our odds of drawing too much ramp instead of action. Let’s pretend for a second that we made the crazy decision to include 25 sources or ramp in our deck. How close would we be to 60-75%

Let’s explore how we can use combinations to calculate our deck’s likelihood to draw any given number of a source in our opening 7. We can plug our numbers into the following equation:

• x = total cards in deck that are not ramp
• a = number of cards in opening hand of the given type (x)
• y = total cards in deck that are ramp
• b = number of cards in opening hand of the given type (y)
• z = total cards in deck
• d = opening hand size (7)

If we plug in to find out the probability of an opening hand with no ramp, we find that chance of not having a single ramp card is 12.09%, meaning there is an 87.91% chance that we'll have a ramp spell in our opener. This is much higher than the 60-75% range we were looking to hit.

This is great, as we do want ramp spells, but we don't want to be drawing more than 1-2 in our opening hand because we need to get to the action in our deck. Let’s take a look at the odds of drawing more than 1 source of ramp.

Odds of 1 ramp spell - 31.11%

Odds of 2 ramp spells - 32.46%

Odds of 3 ramp spells - 17.78%

Odds of 4 ramp spells - 5.51%

When we take a deeper look at the math, we see that in over 20% of games we'll draw 3 or more ramp spells in our opening hand. Ramping is great, but in this deck it's not a key part of our strategy, so we can safely determine that this is too high a rate to see ramp spells at.

Enchantments on the other hand, are key to our deck’s gameplan. If we look at the same odds and apply them to the enchantment portion of our deck, I think that 3+ enchantments in my opening hand 20% of the time is a little lower than I would like. Let’s take a look at how 35 enchantments affects my odds:

Odds of 0 enchantment spells - 4.17%

Odds of 1 enchantment spell - 17.63%

Odds of 2 enchantment spells - 30.47%

Odds of 3 enchantment spells - 27.93%

Odds of 4 enchantment spells - 14.65%

Now these are odds I can get behind. This rate of enchantment presence should mean that both our opening hand and deck will be full of cards that advance our gameplan. This same strategy can be applied to each category in our deck, including creatures and removal. You just have to consider the rate at which you want to draw the given type of card for your strategy and have the numbers match.

The final step is possibly the most important category of our deck, and that's the card advantage that helps us navigate through our key enchantments. Let’s take a look at our deck’s mana curve to determine if we have an appropriate amount of card advantage. Calculating both your deck’s mean (average) and median (middle number) can be helpful in determining if you have sufficient card draw. 

• Your deck’s mean is simply the average mana value. Our deck is coming in just under a mean value of 2.5.
• The median is the value that separates the higher and lower halves of the data. It can be calculated by looking at you mana curve and finding the card in the middle and noting its mana value. Our build has a median value of 2.
  •  Calculating the median is an important tool that can give you a better understanding of where your mana values are concentrated, as outliers in either direction have less of an impact on the median than they do on the mean.

What would happen if you draw 1 card per turn, play 1 land per turn, and cast 1 spell turns 2-4 and start casting multiple spells by turn 5? How quickly will you run low on cards?

2.5 Mean / 2 Median	Turn 1	Turn 2	Turn 3	Turn 4	Turn 5	Turn 6
Cards out of hand	1	2	2	2	3	3
Cards in hand (end of turn)	7	6	5	4	2	0

It’s turn 6 and you are out of cards. A low mana curve helps us efficiently advance our gameplan and cast multiple spells per turn quicker than our opponents. However to support our deck, with its efficient cost and fast rate of play, we must be packing sufficient card advantage.

There are three key elements to focus on when selecting card advantage

• Synergy - How does this card advance my strategy?
• Card Advantage Value (CAV) - How many additional cards end up in your hand
• Deck Velocity Value (DVV) - How quickly do you move through your deck towards key spells. How many cards deep in your deck a spell reaches

Let’s start with synergy. Enchantress’s Presence is an ideal form of card advantage as it triggers off casting our enchantments, generating us consistent value throughout the game.

Omen of the Sea also has great synergy in our deck, but I don't categorize it in the card advantage portion of my deck as it only has a neutral card advantage value (CAV) (+0). It's important to consider how many cards up you are after a given interaction to determine a card’s CAV. A card that has a one-time effect that replaces itself is a cantrip and not card advantage.

Finally, there is DVV. Our deck is comprised of 35 enchantments, which means any given card has a ~35% chance of being an enchantment. A card like Sign in Blood has a CAV of (+1), but it only takes us 2 cards deeper into our deck, giving it a Deck Velocity Value (DVV) of (2). Seeing any 2 random cards in our 99 gives us a 58.4% chance of hitting at least one enchantment. Fact or Fiction, on the other hand, has a variable CAV, but it will always have a DVV of 5. Digging 5 cards deep gives us an 89.3% chance of seeing at least one enchantment.

In recap, we were able to use probability to adjust our spell ratios according to our deck strategy. We ensured that our deck has enough gas to keep drawing action and enough card advantage to prevent our hand from running low. There are no hard and fast ratios to use; you just need to let your strategy dictate your deck construction.

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This article only begins to scratch the surface of how you can utilize mathematical principles, probability, and game theory to improve your strategy and streamline your deck construction. Future articles will take a deeper look at more elements of deck construction as well as the decisions we make in-game. Playing with the odds in your favor will create a much better game experience for yourself where you will be able to participate at all stages in the game. Doing the math is worth it. Trust me, there is no worse feeling than being stuck with an empty hand watching your friends play.