Magic Math – Eldrazi in Modern

Modern is dominated by Eldrazi. But even in the Blind Eternities, math still applies. Today, I’ll offer some probabilities, a few percentages, and other insights on our new Eldrazi overlords. Along the way, you may learn about the useful multivariate hypergeometric distribution, but you can also choose to skip the math and go straight ahead to the conclusions.

Should You Be Scared of the Turn-2 Kill?

The colorless variant that propelled three players into the Top 8 of Pro Tour Oath of the Gatewatch has fallen out of favor recently—it’s weak against most other Eldrazi decks—but it still offers a potential turn-2 kill. As outlined in LSV’s article, it requires the following sequence:

Turn 1

Turn 2

But how often does this happen? To make things concrete, let’s assume that you’re on the play and never mulligan. So you either have to draw the perfect 7-card opening hand or you have to have a near-perfect opening hand followed by the missing Eldrazi Temple, Simian Spirit Guide, or Reality Smasher on your second turn.

Can we expect to pull off this turn-2 kill on the play once every hundred games? Once every thousand games? Ten-thousand games? More?

Probability theory to the rescue!

Multivariate hypgeom dist

Click to enlarge.

With this relatively well-known extension of the hypergeometric distribution, which intuitively can be described as dividing the number of successful combinations by the number of all possible combinations, you can run the numbers. Consider the following 60-card deck:

 

 

OpeningHand prob

Okay, that’s bizarrely small. It means that, in the long run, this will happen approximately once per 370,000 games. To put that number into perspective: In a huge Grand Prix-like event with 8,000 competitors, all of whom play the colorless Eldrazi deck in 15 rounds with 3 games each—none of which is realistic—you would expect to observe this perfect opening hand once.

That said, you don’t have to draw this perfect opening hand to get the turn-2 kill. You can draw Reality Smasher, Eldrazi Temple, or Simian Spirit Guide on turn 2 instead. If you account for that possibility as well, then you get the following.

PutTogether

So that’s still once per approximately 107,000 games in the long run. This means that in our fictive 8,000-player 15-round mono-Eldrazi event, a turn-2 kill on the play would likely happen 3 or 4 times. More realistically, at the upcoming Grand Prix weekend with a Day-2 cutoff, possible 2-game matches, and much less popularity of the colorless Eldrazi deck, I would bet against it happening even once.

So I wouldn’t be scared of the turn-2 kill.

How Would a Ban of Eldrazi Temple or Eye of Ugin Impact the Deck?

 

Since the main power of the Eldrazi deck lies in Eye of Ugin and Eldrazi Temple, I’ve been told by experienced Eldrazi pilots that they tend to mulligan if they don’t have one of those 2-mana lands.

As a result, I was curious about the following question: If you mulligan every hand that doesn’t contain at least one of the 8 Ancient Tombs (and only mulligan those hands), how often do you start with 7, 6, or 5 cards? Before scrolling down, take a guess yourself!

A simple multiplication of hypergeometric probabilities yields the following table. For instance, the probability of keeping a 6-card hand, using horrible Excel notation, is determined as HYPGEOM.DIST(0,7,8,60,FALSE)*(1-HYPGEOM.DIST(0,6,8,60,FALSE)).

Opening hand size after mulligans percentage probability:

Keep 7: 65.4%
Keep 6: 20.6%
Keep 5: 7.4%
Keep 4 or less: 6.7%

Mulligans provide incredible card selection. If an Eldrazi player is only concerned with finding a 2-mana land, then they’ll be able to do so with at least a 6-card opening hand in over 85% of the games. In practice, the number will be a little lower because all-land or one-land hands will still prompt extra mulligans, but this consistency still scares me.

Many people have advocated banning either Eldrazi Temple or Eye of Ugin.

Opening hand size after mulligans percentage probability after a potential ban:

Keep 7: 39.9%
Keep 6: 21.1%
Keep 5: 11.7%
Keep 4 or less: 27.2%

Much less scary. The deck would probably still be playable, but it would take a noticeable drop in 2-mana-land consistency, especially taking into account that the strategy of mulliganing toward a 2-mana land would become foolish.

How Realistic is a turn-2 Thought-Knot Seer on the play?

You’ve seen that the second-turn kill is extremely unlikely. You’ve also seen that finding a 2-mana land is relatively easy. But how about a Thought-Knot Seer on turn 2, which requires a proper combination of two 2-mana lands? That’s not as devastating as a turn-2 kill, but still pretty powerful.

It is possible to analytically determine the probability of having the 4/4 in addition to 2 Eldrazi Temple or, alternatively, 1 Eldrazi Temple and 1 Eye of Ugin on turn 2, but this requires a time-intensive and error-prone enumeration of scenarios. So I made things easier for myself with a quick simulation. In this simulation, I assumed that you mulligan a hand if and only if it doesn’t contain a 2-mana land, that for simplicity you don’t use the free scry, and that you’re always on the play.

The result? A turn-2 Thought-Knot Seer happens in 11.8% of the games. In contrast, the Eldrazi player will hold a Thought-Knot Seer without the mana to play it on turn 2 in 27.0% of the games. What this tells us is that hands with multiple 2-mana lands are relatively uncommon. Phew. It’s still somewhat scary to face a turn-2 Thought-Knot Seer once every 9 games or so against the Eldrazi deck, but it’s not unbeatable.

I’ll be in Bologna this weekend to provide text coverage of the Modern Grand Prix there (which is sold out, by the way) and I hope to see plenty of innovation from brewers who attack the weak spots in the Eldrazi strategies.

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