Dominaria contains several cards I found interesting from a mathematical perspective. I already analyzed Mox Amber here and Wizard’s Lightning, Wizard’s Retort, and Ghitu Lavarunner here, but there are so many more. Let’s run some numbers!

How Often Will You Hit a Certain Type of Card in Your Top 5?

These cards all have a similar effect: Look at the top 5 cards of your library and take a certain type of card (historic, creature, or artifact) from among them. We can find the following hypergeometric probabilities if we remove one of these cards from a 60-card deck and put the card or its ability on the stack.

Hits in deck 8 9 10 11 12 13 14 15
Hit probability 53.1% 57.7% 61.9% 65.8% 69.4% 72.6% 75.6% 78.3%
Hits in deck 16 17 18 19 20 21 22 23
Hit probability 80.8% 83.0% 85.0% 86.9% 88.5% 90.0% 91.3% 92.5%

Weatherlight – So in a 60-card deck containing 20 historic cards, which is the same amount that the “Vehicle Rush” Challenger Deck contains, a Weatherlight trigger will find at least one historic card 88.5% of the time. That is a good level of consistency for a bonus trigger. I would already have been happy with less.

For Mardu Vehicles, Weatherlight could be a decent sideboard option for the grindier post-sideboard games. It can also block Rekindling Phoenix and live. On the other hand, Weatherlight is answered cleanly by Abrade in an unfavorable mana trade. Given the prevalence of Abrade in the current Standard, we may have to wait until the Standard rotation before Weatherlight will see competitive play.

Board the Weatherlight – For Board the Weatherlight, I would recommend at least 21 historic cards, as missing more than 10% of the time is unacceptable. If Board the Weatherlight whiffs, you have missed the boat completely.

In pre-Dominaria Standard, there are not many decks with 21 historic cards. Mardu Vehicles can get there, but as an aggro deck it’s not interested in slow card selection. mono-white midrange decks, even with Oketra’s Monument and Legion’s Landing, aren’t close to 21. The most realistic homes for Board the Weatherlight are weird new combo decks based around historic cards like Teshar, Ancestor’s Apostle. For such decks, it’s worth noting that Inventors’ Fair, the only legendary land in Standard, can be taken with Board the Weatherlight as well.

The Antiquities War – If I put The Antiquities War in my deck, then I’d want to hit an artifact at least once, preferably twice. With 15 artifacts, the above table show that a single trigger has a hit ratio of 78.3%. This means that for two triggers, you’ll hit twice with probability 61.3%, hit once with probability 34.0%, and miss completely only in 4.7% of the cases. Although more artifacts obviously increases your consistency, these numbers are reasonable enough if you plan to abuse the third chapter with non-artifact spells like Servo Exhibition or Hidden Stockpile. So 15 artifacts would be my recommended minimum for The Antiquities War.

In Modern Affinity, I could see The Antiquities War in the anti-midrange sideboard slot traditionally occupied by Bitterblossom, Shapers’ Sanctuary, or Hazoret the Fervent. But the Saga would only be worth it against slow, nonblue decks like Jund or Mardu Pyromancer. Against blue decks that have Cryptic Command to tap your 5/5s, Karn, Scion of Urza, or Tezzeret, Agent of Bolas (if your mana base can afford it) are almost surely better. But depending on the metagame, The Antiquities War is worth testing.

Memorial to Unity – Memorial to Unity is a fine inclusion for mono-green decks built around Steel Leaf Champion. Such decks will surely have a ton of creatures, so you’ll hit almost every time. Back in the day, mono-red decks could afford to run 2-3 tap-lands like Teetering Peaks, Smoldering Spires, and Ghitu Encampment, and I think the same holds for mono-green in the new Standard.

How Many Legends Do You Need for a Legendary Sorcery?

There are three other legendary sorceries in Dominaria (Kamahl’s Druidic Vow, Primevals’ Glorious Rebirth, and Urza’s Ruinous Blast) but they only fit in decks like G/W Legendary that are filled to the brim with legends. For such decks, controlling a legend won’t be a big restriction.

So let’s focus on Jaya’s Immolating Inferno, Karn’s Temporal Sundering, and Yawgmoth’s Vile Offering. These cards don’t get better if you have multiple legends—you only need one. Taking into account that opponents will interact and try to kill some of our legends, I’ll simulate a midgame setting as follows:

  • I consider a 60-deck with 25 lands, four copies of a specific legendary sorcery, and a certain number of legendary creatures or planeswalkers.
  • I mulligan a hand if it contains zero, one, six, or seven lands. In case of a mulligan, I ignore the free scry for simplicity.
  • I assume that I am always able to cast any legend I draw. This assumption is justified for cheap legendary permanents if the mana base contains enough colored sources.
  • I am on the play and on the draw with equal 50-50 probability.
  • I assume that opponents have a certain number of removal spells (0, 1, or 2) that they immediately use to kill any legend we play. After all, in real games of Magic, powerful creatures or planeswalkers rarely go unanswered.
  • I consider the following question: If I have a legendary sorcery in hand on turn 6 and at least five lands, then what is the probability that I started the turn with a legend under my control? In other words, a freshly drawn legend on turn 6 won’t help, and I determine a conditional probability for settings where the cards drawn up until turn 6 contain at least one legendary sorcery and at least five lands. (This corresponds to casting Jaya’s Immolating Inferno for X=3, is a perfect match for Yawgmoth’s Vile Offering, and fits Karn’s Temporal Sundering if we played Baral, Chief of Compliance or a mana ramp spell.)

My simulation gives the following results. If the probabilities are lower than you expected, then remember that I only focus on relevant games where at least five of your cards are taken up by lands.

If your opponent won’t interact at all, then 14 legends is easily more than enough to enable a legendary sorcery. But if your opponent has two removal spells for your legendary creatures or planeswalkers every game, then you may need as many as 26. I believe the truth is somewhere in between.

With 20 legends, you’re over 90% to control at least one legend for a legendary sorcery at the start of turn 6 when your opponent has cast a single Vraska’s Contempt. That sounds reasonable to me, so my recommended minimum is 20 legends for a 60-card deck. As a result, I wouldn’t slot Jaya’s Immolating Inferno into Mono-Red Aggro just because it has Kari Zev and Hazoret. It’s actually pretty hard to make these legendary sorceries work, and the amount of deck building effort required may not be worth it.

For a 40-card deck, the equivalent of 20 legends (for the 60-card deck) is 12 legends. But you don’t need 90% consistency for Jaya’s Immolating Inferno in Limited. A card that is a blank 50% of the time and a one-sided Wrath of God the other 50% of the time would already be fine, and in game 1 your opponent won’t realize how important it will be to kill your legend.

But even taking those things into account, I’d like to have six legends at minimum to support a legendary sorcery in Limited, yielding analogous castable probabilities of 82.9% against zero removal spells and 43.4% against one removal spell. But that’s still a hefty deck building requirement. With zero common legends in the set, it will be tough to gather six of them in Draft and especially Sealed. I fear that at the prerelease, players will put legendary sorceries into way more decks than they should.

How Many Artifacts Do You Need for Zahid, Djinn of the Lamp?

If we replace Yawgmoth’s Vile Offering with Zahid, Djinn of the Lamp, replace “legendary creature or planeswalker” with “artifact,” and consider turn 4 instead of turn 6 with the same type of analysis, we get the following:

Given that artifacts won’t die as easily as legends, I believe that 17 artifacts are a good target. This matches the number of artifacts I used to run in Mardu Vehicles, and it’s in line with the numbers that Paulo Vitor chose for his Zahid brews. If opponents have half a removal spell on average, then with 17 artifacts you’ll be able to cast Zahid on turn 4 somewhere between 80% and 90% of the time.

That said, opponents are less likely to spend removal spells on Prophetic Prism than on Foundry Inspector, so the minimum really depends on the types of artifacts you have in your deck. In the mono-Prophetic-Prism deck, 13-14 artifacts would already be fine.

How Many Cards Can You Expect to Get with Demonlord Belzenlok?

Suppose you cast Demonlord Belzenlok when your deck contains 27 nonland cards (which is what you can expect by turn 6). Varying the number of four or more mana cards remaining, we can use basic probability theory to determine the distribution and expected number of cards granted by Professor Belzenlok.

Number of 4+ CMC cards among 27 nonland cards 1 card 2 cards 3 cards 4 cards 5+ cards Expected number
6 77.8% 17.9% 3.6% 0.6% 0.1% 1.27 cards
7 74.1% 19.9% 4.8% 1.0% 0.2% 1.33 cards
8 70.4% 21.7% 6.1% 1.5% 0.4% 1.40 cards
9 66.7% 23.1% 7.4% 2.2% 0.7% 1.47 cards
10 63.0% 24.2% 8.7% 2.9% 1.2% 1.56 cards
11 59.3% 25.1% 10.0% 3.8% 1.9% 1.65 cards
12 55.6% 25.6% 11.3% 4.7% 2.8% 1.75 cards
13 51.9% 25.9% 12.4% 5.7% 4.1% 1.87 cards
14 48.1% 25.9% 13.5% 6.7% 5.7% 2.00 cards
15 44.4% 25.6% 14.4% 7.8% 7.8% 2.15 cards
16 40.7% 25.1% 15.0% 8.8% 10.4% 2.33 cards
17 37.0% 24.2% 15.5% 9.7% 13.6% 2.55 cards
18 33.3% 23.1% 15.7% 10.5% 17.4% 2.80 cards
19 29.6% 21.7% 15.6% 11.0% 22.1% 3.11 cards
20 25.9% 19.9% 15.2% 11.4% 27.6% 3.50 cards
21 22.2% 17.9% 14.4% 11.4% 34.1% 4.00 cards
22 18.5% 15.7% 13.2% 11.0% 41.7% 4.67 cards
23 14.8% 13.1% 11.5% 10.1% 50.5% 5.60 cards
24 11.1% 10.3% 9.4% 8.6% 60.5% 6.97 cards

As an example of how to read this table: If 10 out of those 27 cards had a converted mana cost of 4 or greater, then you would merely get one card in 63.0% of the cases, you’d get exactly two cards with probability 24.2%, exactly three cards with probability 8.7%, and so on. The expected number of cards you’d receive in this scenario is 1.56.

Unless over half your deck is filled with 4+ mana cards, you can’t expect to get more than two cards out of Belzenlok. Since normal decks can’t afford such a skewed mana curve and given that the enters-the-battlefield triggers of Demon of Dark Schemes or Noxious Gearhulk are arguably even better than a supercharged Night’s Whisper, I don’t expect to see many Demonlord Belzenloks in normal decks.

But Liliana’s end boss may fit in a deck with plenty of aftermath cards. Cut // Ribbons, Never // Return, and Spring // Mind give you something to do early on while having a converted mana cost of 4 or more. Taking this idea to the extreme, you could draw about five cards on average in a deck where 29 out of your 35 spells have a converted mana cost of 4 or greater (as this ratio corresponds to about 22 or 23 spells out of 27 in the above table).

But what if we could guarantee the exact set of cards we get? Allow me to introduce a horrendously convoluted contraption.

Standard Belzenlok Brew

This deck is not even close to competitive. And with that, I mean it’s utterly, utterly unplayable. But it has a clear plan:

Turn 2: Cast Board the Weatherlight, hoping that the top 5 contains four lands and another copy of Board the Weatherlight. (Even assuming we hold one Board the Weatherlight, hitting another happens in only 4% of the cases, so the whole combo plan already fails 96% of the time.) Reveal nothing, push the lands to the bottom, and put Board the Weatherlight fifth from the bottom.

Turn 6: Cast Demonlord Belzenlok. This part of the plan is at least somewhat consistent, since you’re almost guaranteed to hit your land drops in a 46-land deck, and you’re over 60% to naturally draw Demonlord Belzenlok by turn 6. Assuming that we started the turn at a double-digit life total (completely realistic after not doing anything for the first five turns…) we repeat the enters-the-battlefield process until we hold every nonland card in our deck and our library consists of 4 lands. (We needed that Board the Weatherlight as a “stop” button, otherwise we’d be decked on the next turn.) Discard all excess copies of Demonlord Belzenlok, the extra Board the Weatherlight, and one Karn’s Temporal Sundering. Keep two Karn’s Temporal Sundering, two Brass’s Bounty, one Anointed Procession, one Metallurgic Summonings, and one Revel in Riches in hand.

Turn 7: Since our opponent wasn’t able to kill us yet and obviously lacked a removal spell for our Demon, we untap safely. We draw a land (at least that’s guaranteed), play it, and cast Karn’s Temporal Sundering.

Turn 8: We draw and play a land and cast Karn’s Temporal Sundering. Our hand is now two Brass’s Bounty, one Anointed Procession, one Metallurgic Summonings, and one Revel in Riches.

Turn 9: We draw and play a land. We tap all our lands for mana, cast Brass’s Bounty, and sacrifice all our Treasures to get 11 mana total. We then cast Anointed Procession and Brass’s Bounty to yield 18 Treasures. Next, we play and exile Metallurgic Summonings (8 Treasures remaining) to return 2 Brass’s Bounty and 1 Karn’s Temporal Sundering (which we cleverly discarded to hand size on turn 6). We go up to 30 Treasures, cast Revel in Riches and Karn’s Temporal Sundering, and pass the turn with 19 Treasures in play.

Turn 10: At the beginning of our upkeep, we win the game!

You may decide where this deck ranks on a scale from next-level nonsense to worst idea ever.