A commonly used mechanic is to look at the top cards of your library and take a card that satisfies a certain characteristic. Good examples are Glint-Nest Crane, Board the Weatherlight, and Ancient Stirrings, and Core Set 2019 introduces several new ones. Since you have to carefully build your deck around cards like these, it’s useful to know the likelihood of hitting.

# Hypergeometric Probabilities

Let’s consider Glint-Nest Crane. The probability of having at least one artifact in your top 4 cards obviously depends on the size and contents of your library at the time you cast it. For example, if you have 21 artifacts and 38 non-artifacts remaining in a 59-card library—without worrying how you got to that spot—then the first card you see is not an artifact with probability 38/59 or 64.4%. If it indeed was a non-artifact, then the second card is not an artifact either with probability 37/58 or 63.8%. For the third card, it’s 36/57 or 63.2%. For the fourth card, it’s 35/56 or 62.5%. Multiplying all four probabilities together, we get a miss probability of 16.2%, which translates to a hit probability of 83.8%.

As I mentioned, this number depends on the size and contents of your library when casting Glint-Nest Crane. I could provide a probability table for every possible size-content combination, but that would be overwhelming and devoid of practical use. For deck building purposes, we just want one number for each deck: the expected hit probability when casting Glint-Nest Crane in a game. Using these numbers, we can check the minimum number of artifacts required for an expected hit probability of, say, 83%.

There are various ways to define and model this notion of “casting Glint-Nest Crane in a game.” We could factor in mulligans, condition on drawing certain combinations of cards, assume a certain turn of the game, and so on. Such complications may be worth it for Daring Buccaneer, but for cards like Glint-Nest Crane I find it more appealing to simply remove one copy from the deck, draw any number of cards (representing my opening hand and any number of draw steps), and finally put its ability on the stack. The cards drawn don’t affect the hit probability, as long as there are still four cards left in the library for Glint-Nest Crane. I don’t condition on drawing at least two lands because the probability of having enough mana to cast Glint-Nest Crane is both extremely high (and thus its influence on our library is negligible) and increasing every turn. The benefit of my approach is that it stays applicable throughout the entire game.

In mathematical terms, I’m taking the hypergeometric probability of seeing at least one success in a certain number of draws from a 59-card deck with a given number of successes. The corresponding hit probabilities are shown below.

Click to enlarge.

As an example to illustrate the use of this table, suppose you have a 60-card deck with 21 artifacts and four Glint-Nest Cranes. According to the entry in the “Top 4 cards” row and the “21 hits” column, you can expect an 83.8% hit probability when casting Glint-Nest Crane in a game. The underlying calculation was explained in this section’s opening paragraph.

# Application to M19 Cards

### Vivien Reid

The +1 ability of this new green planeswalker will almost always hit something, but you won’t get a creature every single time. Even in a 26-creature deck, the table indicates that Vivien Reid is “only” 91.0% to hit a creature. That’s still fine, and definitely better than drawing a random card in the late game.

Her other abilities are unexciting. Destroying Heart of Kiran, Cast Out, or Glorybringer is useful but narrow. Against many decks, Vivien Reid won’t be able to protect herself the turn she comes down, which is an issue for a 5-mana planeswalker. The ultimate is hard to reach and reliant on having creatures on the battlefield.

The most likely use for Vivien Reid is in the sideboard of green midrange decks, as a hard-to-answer card advantage engine against control. One advantage that Vivien Reid has over Nissa, Vital Force is that she can provide an out to Lyra Dawnbringer in post-board games.

### Militia Bugler

Imperial Recruiter this is not. Since you cannot rely on getting the creature you want—you’re only 25.1% to hit a certain 4-of in your top 4 cards—Militia Bugler is more akin to a Rogue Refiner or Carven Caryatid. I would want to have at least 15 hits in my deck before I would consider adding Militia Bugler, and even then the hit probability is still only 70.2%.

So what kind of decks could run this many low-power creatures? Aggro decks might, but they mainly want to play these creatures as early as possible. Adanto Vanguard, for example, is great on turn 2, but it’s unimpressive on turn 4. Also, Militia Bugler doesn’t add a whole lot of power to the board on turn 3, so it doesn’t fit an aggro plan.

I believe that the best home for Militia Bugler is in a combo deck with lots of low-power creatures, including at least one key 4-drop or 5-drop creature. For such decks, Militia Bugler can increase consistency while providing a blocker against aggro. For example, the W/U Gift deck that runs Minister of Inquiries, Champion of Wits, and Angel of Invention (and sometimes Sunscourge Champion or Walking Ballista) might consider a few copies of Militia Bugler. The card could also help historic decks based around Aviary Mechanic and Teshar, Ancestor’s Apostle (and sometimes Walking Ballista and Yahenni, Undying Partisan).

### Elvish Rejuvenator

Hey, it’s a new take on Wood Elves! In a 26-land deck, Elvish Rejuvenator will hit a land 95.3% of the time, which is an excellent level of consistency. Sure, you’ll whiff once per 21 games on average, but Elvish Rejuvenator still seems better than Beneath the Sands and Grow from the Ashes. After all, you get a 1/1 chump blocker. You can take any land as opposed to just basics, and you speed up games by not having to go through the hassle of a midgame shuffle.

For a B/G Ramp deck, Elvish Rejuvenator seems roughly on par with Gift of Paradise, as both buy time against aggro and ramp you by one. To figure out which is best, you mainly have to weigh the benefit of potentially choosing a utility land or Desert from your top 5 against the downside of slightly lower consistency. It depends on your mana base, but I have high hopes for Elvish Rejuvenator.

### Vivien’s Invocation

When paying 7 mana for an effect that might not produce any value at all, I want a good amount of consistency—definitely not less than 90% and probably closer to the 95% territory. Looking at the table, this requires 16-20 creatures.

By itself that’s a reasonable ask, but to get some value out of Vivien’s Invocation, these creatures should be big. I don’t want to pay 7 mana for a random Grizzly Bears that deals 2 damage. And I have no clue how you’re building a Standard deck with 7 mana sorceries and many big creatures.

Even then, what is the best-case scenario? Zetalpa, Primal Dawn? It will just get this dealt with by Vraska’s Contempt or Settle the Wreckage. Polyraptor? Sounds like fun, but I’m just not seeing the competitive viability. Vivien’s Invocation is just 1 mana too expensive.

### Liliana, Untouched by Death

In a deck with 22 Zombies, which is typical for such a tribal deck, you’re 76.1% to hit a Zombie in your top three cards. That’s an expected life swing of 3, which is better than Chandra, Torch of Defiance’s first ability on turn 4. Either way, Liliana’s power mainly lies in her minus abilities, which are a good fit for any Zombie tribal deck.

Andrea Mengucci recently wrote about Zombies in Core Set 2019 Standard, so I recommend checking out his article if you’re interested in that archetype.

### Tezzeret

If only this was the Tezzeret included in M19, then I could tell you that it would hit an artifact 90.0% of the time in the 21-artifact U/B Improvise deck, just like The Antiquities War. Unfortunately, M19 instead includes Tezzeret, Artifice Master, so I’ll just wrap up here.

In this article, I provided a probability table and applied the numbers to evaluate several M19 cards with a “look at the top X cards of your library and reveal a Y” effect. If M19 is any indication, then we’ll keep seeing this mechanic in increasing numbers in future sets. The table in this article can remain a handy reference to evaluate such new cards.