*Ravnica Allegiance *contains several cards I find interesting from a mathematical perspective. Let’s run some numbers and build some decks!

**You Need 26 Creatures to Hit Over 1.5 on Average with the New Domri**

That’s an above-average starting loyalty for a 4-mana planeswalker.

The most common sequence will be to start with an immediate activation of Domri’s -3 ability. It can yield a decent amount of card advantage, but only if your deck contains enough creatures. Below, you can find the numbers for the case where you remove one Domri, Chaos Bringer from the deck, blindly draw any number of cards, and then put the ability on the stack.

The first column represents the number of creatures in your original 60-card deck. The next three columns describe the probability distribution for the number of creatures you will see in your top four. The final column shows the expected number of creatures you can put in your hand.

Creatures | P(0 hits) | P(1 hits) | P(2+ hits) | E[creatures] |

16 | 27.1% | 43.4% | 29.5% | 1.02 |

17 | 24.6% | 42.9% | 32.5% | 1.08 |

18 | 22.3% | 42.2% | 35.6% | 1.13 |

19 | 20.1% | 41.2% | 38.7% | 1.19 |

20 | 18.1% | 40.2% | 41.8% | 1.24 |

21 | 16.2% | 38.9% | 44.9% | 1.29 |

22 | 14.5% | 37.6% | 47.9% | 1.33 |

23 | 12.9% | 36.1% | 51.0% | 1.38 |

24 | 11.5% | 34.5% | 54.0% | 1.42 |

25 | 10.2% | 32.9% | 56.9% | 1.47 |

26 | 9.0% | 31.2% | 59.8% | 1.51 |

27 | 7.9% | 29.4% | 62.7% | 1.55 |

28 | 6.9% | 27.7% | 65.4% | 1.59 |

29 | 6.0% | 25.9% | 68.1% | 1.62 |

30 | 5.2% | 24.1% | 70.7% | 1.65 |

31 | 4.5% | 22.3% | 73.2% | 1.69 |

32 | 3.9% | 20.6% | 75.6% | 1.72 |

I wouldn’t put too many noncreature spells in a Domri deck. Here’s a first list using 28 creatures, where Domri would yield 1.59 creatures in expectation:

**Standard Gruul Riot**

The riot dished out by Domri’s +1 and Rhythm of the Wild—appropriately named “They call me Mister Pig!” by Scryfall when the actual card name wasn’t known yet—improves Ghalta and Kraul Harpooner substantially. With haste, they may be able to attack for 12 before the opponent can answer them with sorcery-speed removal.

Growth-Chamber Guardian also benefits from riot because you get to search for another Crab right away when it enters with a +1/+1 counter. You can easily build a squadron of Watchcrabs with this deck.

**The Expected Number of Tokens Created by Mirror March is One**

The expected number of tokens you get from a single trigger can be expressed as an infinite sum. It evaluates to one:

So Mirror March is like Doubling Season for creatures, albeit with more variance attached.

Really want to play Mirror March, play a Fanatical Firebrand and win 20 flips in a row on video. The math is pretty dubious though.

— Saffron Olive (@SaffronOlive) January 9, 2019

Well, the probability of winning 20 flips in a row is 0.5 to the power 20, which is a 1 in a million chance. (To be more precise, it’s 1 in 1,048,576.) A dubious plan indeed.

**Mass Manipulation is Powerful, Yet Difficult to Cast**

My updated colored sources article never considered 2UUUU spells, but I would recommend 19 blue sources if you would like to support it as a 6-drop in a 60-card deck. (This would give you a 94.9% probability to have 4 blue sources by turn 6 on the play, conditional on drawing at least 6 lands by turn 6 in a 24-land deck, after mulligans.)

Weirdly enough, the most likely home for Mass Manipulation might be in mono-black. That deck can use Cabal Stronghold to fuel large X-values and Chromatic Lantern to provide 4 blue.

In Limited, quadruple blue is a lot, especially for traditional 9-8 or 10-7 mana bases. The equivalent probabilities of casting Mass Manipulation on curve can be found in the table below.

Blue sources in 40 cards | P(cast turn 6) | P(cast turn 8) |

9 | 47.9% | 80.1% |

10 | 62.1% | 90.7% |

11 | 75.2% | 96.7% |

12 | 86.0% | 99.3% |

If I draft Mass Manipulation, then I’d try to go nearly mono-blue so that I can reasonably run 11 or more Islands and cast it consistently as a 6-drop. Otherwise, I would aim for a 10-7 mana base where I would evaluate the XXUUUU spell as a super-powerful 8-drop that I can also cast for half its effect 62.1% of the time when I have 6 lands on turn 6.

**Esper Control Mana Bases Have Gotten Challenging**

These two cards are substantial boons to control players. Kaya’s Wrath is a full mana cheaper than Cleansing Nova. And unlike Deafening Clarion, it can mop up Venerated Loxodon, Carnage Tyrant, and Crackling Drake. Absorb will be better than Sinister Sabotage most of the time, especially when Banefire is around. Just compare Revitalize to Opt to see that gaining 3 life is worth more mana than scrying 1.

To consistently cast Absorb on curve in a 26-land deck with 8 blue-white dual lands, the other 18 lands need to be 11 blue lands and 7 white lands. (This would give you a 91.0% probability to have WUU mana by turn 3 on the play, conditional on drawing at least 3 lands by turn 3 in a 60-card deck, after mulligans.)

To come even close to consistently casting Kaya’s Wrath on curve in a 26-land deck with 8 white-black dual lands, the other 18 lands need to be 9 white lands and 9 black lands. (This would give you a 89.4% probability to have WWBB mana by turn 4 on the play, conditional on drawing at least 4 lands by turn 4 in a 60-card deck, after mulligans. This is slightly below my desired level of consistency, but barely acceptable if need be.)

Since 3-color decks in *Ravnica Allegiance* Standard could run up to 12 shocklands and 12 checklands, it is possible to build a mana base that can either play Absorb on turn 3 or Kaya’s Wrath on turn 4. Doing both, however, is a challenge, especially since basic Island in a deck with a WWBB spell is a liability and should be avoided. The following mana base comes close, missing by only one blue source, but it’s not pretty:

That’s a **lot** of taplands. The Guildgates in particular look abysmal. Fortunately, there are several alternatives. You could replace the Guildgates with Plains and Swamp while adding card selection spells like Opt to smooth out your draws. Or you could simply accept that you can’t cast Absorb on turn 3 every game. Yet my favored solution would be to merely splash blue and use Syncopate for countermagic instead. Such a splash-blue deck could also turn a shockland and a checkland into two more basics, which reduces tapland worries.

**Wilderness Reclamation Makes it Easier to Take Infinite Turns**

Suppose you control Teferi, Wilderness Reclamation, a transformed Azcanta, and 6 other untapped lands as you move to your end step. This means that you can activate Azcanta twice and still have enough mana left over to cast Nexus of Fate. So if you’re casting Nexus of Fate, Wilderness Reclamation effectively produces two additional Azcanta activations in this spot, which is absurd.

But if your deck contained four Nexus of Fate and 40 blanks at this point, what is the probability that you can take **all** the turns?

As it turns out, this probability is 19.9%, which is not all that high. Yet it would be 47.6% if you already held a Nexus at the start of the turn, 45.2% if the number of non-Nexus cards in their library is only 30, and 79.2% with both adjustments. These numbers were determined by adapting the approach from this article.

These probabilities only represent a lower bound because the 40 non-Nexus cards aren’t actually blanks. I don’t advocate conceding right away against a Turbo Fog player in this spot—I would give them at least a few turns to potentially fizzle—but the likelihood that you could get out of the stranglehold is small.

**Rakdos is Indeed a Showstopper**

With X total creatures on the battlefield (excluding Demons, Devils, and Imps) the probability that Rakdos destroys all opposing ones but spares all of yours is 0.5 to the power X. So if both players control four creatures, then you’re 1 in 256 to get the perfect one-sided sweep. You may share your excitement on social media at that point.

The Thanos probability of killing exactly 50% of all the creatures on the battlefield is 0.500 for X=2, 0.375 for X=4, 0.313 for X=6, and 0.273 for X=8. I can only hope that Rakdos has a nice finger-snapping animation on *MTG Arena*.

**You Need 21 Creatures to Hit 90% of the Time with Incubation**

With 21 creatures in 60 cards, you’re 90% to hit at least one in your top 5, based on the table from this article. I wouldn’t like to run fewer than that, as missing more than 10% of the time is a huge downside. And honestly, 90% might still be too low of a floor. Depending on the deck, you might want the 95% consistency corresponding to 26 creatures.

The split card can be particularly useful for combo decks that need to find specific creatures, such as Devoted Druid. Recently, there were four copies of Commune with the Gods in the Top 8 of Grand Prix Portland, found in Steven Ricken’s Abzan Evolution deck. In a deck with Noble Hierarch and Birds of Paradise, Incubation // Incongruity is a strict upgrade to Commune with the Gods. But the best home for Incubation // Incongruity might be the following deck.

**Modern Bandit Lord Combo**

**__matsugan, 5-0 at a Competitive Modern League in June 2017**

This ultra-fast take on the Devoted Druid + Vizier of Remedies combo, designed by Japanese deck building genius Atsushi Ito, can win as early as turn 2 thanks to Hall of the Bandit Lord. The deck already ran four Commune with Nature, but Incubation // Incongruity is much better because it also counts as an instant for Traverse the Ulvenwald.

**High Alert Allows Us to Put the “Rule of 8” into Practice**

There is a huge difference in terms of consistency between playing 4 copies and 8 copies of some effect. High Alert is a good example of this, as it can act as Arcades, the Strategist number 5 though 8.

As a general deck building guideline, if your strategy relies on a certain effect and you have no card selection, then you need at least eight virtual copies. I call this the “Rule of 8.” If you only have four copies of some effect, then based on the tables in this article, you’re only 52.8% to draw at least one by turn 4 on the play. With eight copies, this increases to 79.0%. That’s the difference between a casual brew and a competitive deck.

As a result, the following strategy might become competitively viable.

**Standard Bant Defend**

I’m looking forward to seeing this in action.

Electrodominance is another application of the Rule of 8, as it’s a huge upgrade for Modern decks that try to cast Restore Balance, Ancestral Vision, Wheel of Fate, and/or Living End. Previously, they only had four copies of As Foretold to cast these spells, but now they can have eight copies of that type of effect. Again, moving from 52.8% to 79.0% is a massive difference.

Also, as Grand Prix Prague champion Pascal Vieren suggested to me, there’s a new turn-1 kill in Modern:

- Mountain, Faithless Looting
- Discard Kiki-Jiki, Mirror Breaker and Deceiver Exarch
- Exile two Simian Spirit Guides
- Cast Electrodominance for 0, then cast Living End.

The likelihood of pulling off this sequence on turn 1 is a paltry 0.03%, but I’m a dreamer.

**Sphinx of Foresight Allows You to Reduce Your Land Count**

To figure out if Sphinx of Foresight may allow you to cut some lands from your deck, I will determine the probability of hitting land drops on turns 1, 2, and 3 for varying numbers of lands and Sphinxes in a 60-card deck. I will assume the following mulligan strategy, which is reasonable for a wide range of decks:

- For 7-card openers, mulligan any hand with 0, 6, or 7 lands. Mulligan a 1-lander if and only if it contains 0 Sphinx.
- For 6-card openers, mulligan any hand with 0, 5, or 6 lands. Mulligan a 1-lander if and only if it contains 0 Sphinx.
- For 5-card openers, mulligan any hand with 5 lands. Mulligan a 0-lander if and only if it contains 0 Sphinx.
- Keep any 4-card hand.
- With a mulligan scry and/or any number of Sphinx triggers, we always scry a land to the top and a spell to the bottom. The Sphinx triggers occur after the mulligan scry.
- We are on the play 50% of the time and on the draw 50% of the time.
- We don’t use
*MTG Arena’s*opening hand algorithm, whatever it even is.

Under these assumptions, the table below describes the probability of hitting your first three land drops. The numbers from the first column match the ones from this article, and I adapted the method described there to run the numbers with Sphinx of Foresight into the mix.

Lands | 0 Sphinx | 1 Sphinx | 2 Sphinx | 3 Sphinx | 4 Sphinx |

19 | 80.1% | 80.8% | 81.6% | 82.3% | 83.1% |

20 | 82.9% | 83.6% | 84.3% | 85.1% | 85.8% |

21 | 85.4% | 86.1% | 86.7% | 87.4% | 88.1% |

22 | 87.6% | 88.2% | 88.8% | 89.5% | 90.1% |

23 | 89.5% | 90.1% | 90.6% | 91.2% | 91.8% |

24 | 91.1% | 91.6% | 92.2% | 92.7% | 93.3% |

25 | 92.5% | 92.0% | 93.5% | 94.0% | 94.5% |

26 | 93.7% | 94.2% | 94.6% | 95.1% | 95.5% |

So if all we care about is hitting our first three land drops, then a deck with 23 lands and 0 Sphinx is roughly equivalent to a deck with 22 lands and 3 Sphinx. In other words, if you add 3 Sphinx, then you can go from 23 to 22 lands while retaining the same probability of hitting your first three land drops. You need 4 Sphinx to go from 20 to 19 lands, and you need 2 Sphinx to go from 26 to 25 lands.

The natural home for Sphinx of Foresight is Mono-Blue Tempo. This deck is geared around the power of a single card: Curious Obsession. 0-mana card selection will increase the number of perfect draws, and the fail case of a 4/4 flyer for 4 mana isn’t super embarrassing. Here is how I would build it, partly inspired by whyjesse’s Mythic Reddit post:

**Standard Mono-Blue Tempo**

In this deck, you need to mulligan aggressively, so I’ll amend the the mulligan strategy described above by adding the stipulation that you mulligan every 7-card opening hand without either Curious Obsession or Sphinx of Foresight. If you use this mulligan strategy and scry everything other than Curious Obsession to the bottom, then you will have Curious Obsession by turn 2 61.4% of the time. It would be 59.0% in a 0 Sphinx deck. Overall, the addition of three Sphinx represents a small, yet relevant, consistency boost. The math favors the Sphinxes.

Thanks for reading, and enjoy *Ravnica Allegiance*‘s upcoming release!

## Discussion