# How Reliable Is Hollow One?

Black-Red Hollow One had its breakout performance at Pro Tour Rivals of Ixalan. Taking advantage of Burning Inquiry and Goblin Lore, the deck is capable of cranking out Hollow Ones as early as turn 1. Team Musashi’s Ken Yukuhiro piloted the following build to success at the Pro Tour.

# B/R Hollow One

### Ken Yukuhiro, Top 4 at Pro Tour Rivals of Ixalan

Ken Yukuhiro never disappoints with his innovative deck choices. He’s found success in the past with Tukatongue Thallid, Bone Picker, and Lone Rider, and now we can add Hollow One to that list as well. What’s more, no matter whether he wins or loses, he always has a jovial smile on his face. I will strongly consider him for my Hall of Fame ballot later this year.

But can we quantify how lucky he was here?

On turn 2, with 50 cards remaining in his deck, Ken Yukuhiro casts Goblin Lore. At that point, his remaining 5-card hand contained 1 Hollow One. After drawing 4 more cards, 3 had to be discarded at random, and the commentators were shouting “Not the Hollow One!” After the dust settled, Yukuhiro’s remaining 6-card hand contained not just the Hollow One, but the Hollow Two!

Let’s calculate the exact likelihood of this outcome.

For this, we have to add up the following convolutions of hypergeometric probabilities:

• The probability of drawing 1 Hollow One in 4 cards (from a 50-card deck containing 3 Hollow Ones) multiplied by the probability of finding 2 Hollow Ones in a 6-card sample (out of a population of 9 cards, 2 of which are Hollow Ones).
• The probability of drawing 2 Hollow Ones in 4 cards (from a 50-card deck containing 3 Hollow Ones) multiplied by the probability of finding 2 Hollow Ones in a 6-card sample (out of a population of 9 cards, 3 of which are Hollow Ones)
• The probability of drawing 3 Hollow Ones in 4 cards (from a 50-card deck containing 3 Hollow Ones) multiplied by the probability of finding 2 Hollow Ones in a 6-card sample (out of a population of 9 cards, 4 of which are Hollow Ones)

The result is 9.6%.

While it was much more likely for Yukuhiro to end up with a 0 Hollow Ones (27.6%) or a single Hollow One (62.5%), it’s still well in the realm of possibility to end up with two. If he had gone up to 3 or 4 Hollow Ones, then that would have been truly spectacular, but there’s only a 0.3% chance of that happening.

You can find the full numbers for different starting and ending numbers of Hollow One in the tables below. Each column corresponds to a different hand size. Given that Ken Yukuhiro had 5 cards in hand after putting Goblin Lore on the stack, you can find the above-described percentages in the column “5 cards in hand” and in the rows corresponding to “1 Hollow One in hand before Goblin Lore.”

The tables for both Goblin Lore and Burning Inquiry are based on a 50-card library. But the probabilities won’t change much if you cast the draw-and-discard spells when you have, say, a 48-card or 52-card library. The numbers in the tables will remain a good approximation.

While the odds for specific situations are useful to know, they don’t immediately help determine the overall consistency of the strategy. So let’s move to the big question.

How often will you get double Hollow One on turn 1?

If you thought that 8 power on turn 2 was unfair, then how about two 4/4s on turn 1?

Poor Reid.

But two 4/4s on turn 1 in a single match has to be a rare occurrence, right?

Well, I ran the numbers. I set up a simulation, replicated ten million games, and put the results in a tasty pie chart.

To set up the simulation, which alternates between playing first and drawing first, I had to make a bunch of assumptions on mulligan strategy and gameplay decisions. I told my computer to mulligan a hand if it contains 0, 1, 6, or 7 lands and to also mulligan a 6-card or 7-card hand if it lacks a card-draw spell or cycler. Further, I programmed the gameplay logic for each turn to sequence spells in a way that usually maximizes the upside potential of casting as many Hollow Ones as possible. This meant playing Burning Inquiry over Faithless Looting or a 1-mana Hollow One, and ignoring all other spells. In case we weren’t holding excess spells or lands when resolving Faithless Looting, we’d discard Goblin Lore, Faithless Looting, a possible second land, Burning Inquiry, and Street Wraith, in that order.

To get a feel for the gameplay decisions, here is the bulk of the turn-2 logic transcribed into English:

IF, after cycling Street Wraiths and casting free Hollow Ones, you have at least 2 Faithless Looting/Burning Inquiry combined or no castable Goblin Lores:

THEN play your first land if you missed your land drop on turn 1 but drew one just now, then Inquiry if possible, else Looting if possible, followed by Street Wraith cycling, then cast Hollow Ones, then play a land if possible, take a rainbow to the stars, Inquiry if possible, else Looting if possible, followed by Street Wraith cycling, then cast Hollow Ones.

ELSE, play a land if possible, cast Hollow Ones, then Goblin Lore if possible, followed by Street Wraith cycling and casting of Hollow Ones. Failing that, cycle Hollow One in the hope of drawing into a chain of Street Wraiths to cast a second One for free.

To be fair, you have to have a very high IQ to understand Hollow One. The strategy is extremely subtle, and without a solid grasp of quantum mechanics, most of the lines will go over a typical player’s head.

More seriously, my code is a spaghetti mess that was not designed for cleanliness or readability, but I am reasonably confident the results are valid because they are close to approximations derived via combinations of the probabilities in the tables above.

Let’s move to the results.

What this means is that, under the gameplay/mulligan logic I assumed, the probability of dumping exactly 2 Hollow Ones on turn 1 is 2.5%. In other words, it’ll happen once per 40 games on average. What’s more likely is a single Hollow One on turn 1, which happens 16.6% of the games. The jackpot of 3 or 4 Hollow Ones on turn 1 happens in only 0.2% of the games. All in all, you’re 19.3% to cast at least 1 Hollow One on turn 1.

The remaining 80.7% of the games is split up between 16.8%+2.7%+0.2%=19.7% where, despite missing on turn 1, you manage to cast at least 1 Hollow One on turn 2, and 61.0% where you fail to play an early Hollow One on turns 1 or 2.

# Conclusion

Blurting multiple Hollow Ones onto the battlefield on turn 1 is insane, but it doesn’t happen with regularity—only 2.5% of the time, or once per 40 games on average. Ken Yukuhiro’s quarterfinal against Reid Duke was a statistical oddity, and in my view the deck is not inherently overpowered.

Yet it is reasonably consistent: According to my analysis, you can expect to play at least 1 Hollow One by turn 2 in about 39% of the games. That’s very similar to the rate at which an Affinity player starts with Mox Opal in their 7-card opening hand. Those Mox Opal starts are powerful, just like a 0-mana 4/4 on turn 1 or 2, but don’t happen reliably enough to dominate the format, and they require you to craft your entire deck around them. In that sense, I believe that R/B Hollow One has the perfect power level for Modern.

That said, this deck is not for the faint of heart—you need to be able to handle the high-variance swings. If you accidentally discard Hollow One, you can’t let it break your heart. You have to embrace the RNG.