HearthMath: Making Curves

Introduction

Hey, guys. It’s RaFive with more tips on bending the math of Hearthstone to your advantage!

Last time, we explored how a deeper understanding of the mathematical probabilities of card draw can help you draft and mulligan better. Today we’re delving a bit deeper into the general considerations that should inform the way you put a deck’s mana curve together past the first couple of turns.

If you haven’t read my previous piece in this series, I strongly recommend you do so before proceeding further, since I won’t be going over too much old ground and will be assuming you’ve read and understood the basic concepts previously enumerated. We’re still going over the thought processes you want to employ to approach your deckbuilding more mathematically, so this still isn’t a terribly math-heavy article, but it’s still helpful to understand the basic arithmetic of the thing.

I’d also like to take this opportunity to unveil a HUGE bonus for my fellow HearthStoners — a complete, color-coded hypergeometric probability table specially calibrated for Hearthstone! I’ve done all the math and formatting so you don’t have to. Now whenever you wonder, “what are my odds of drawing into Ragnaros the Firelord by turn 8,” you need wonder no longer — just look at the handy-dandy chart and see that your chances are 26.67%. I’ll be making casual reference to this chart in the article, so you may want to keep it in a tab for easy reference.

As with last time, I’m sharing a decklist for occasional reference as necessary. This one will be a Xixo-ish Imp-losion Zoo list I’ve been running lately (the only major difference is a lack of Sea Giant).

Types of Important Draws

This article has actually taken me quite a bit longer than usual to put together, and it’s not merely all the math. It’s also because I’ve been doing a lot of reading on reddit and HearthPwn to see how people typically talk about drafting. I’ve discovered that most folks draft pretty thoughtlessly, aping netdecks or simply asking themselves what “good,” “popular,” or “synergistic” cards they can fit in at a particular mana cost.

This isn’t the way to do it if you want to make your own decks and succeed (or properly tech netdecks to improve your winrate on ladder). In order to improve your drafting, you need to making fairly fine distinctions between various cards you can pick. The point of drafting a particular mana curve is to ensure you draw particular types or costs of cards by particular turns so that you’re making solid plays at all the important spots. Let’s begin, then, by thinking about what might constitute an “important” draw.

I’d like to propose that there are three basic and overlapping types of “important” draw in Hearthstone, into which almost every card/strategy can be fit: hard win conditions, curve-critical cards, and tech cards. Remember, as we discussed last time, “on curve” means when you play the card, rather than how much it costs. As such, it’s critical you understand the differences between types of important draws, in order to figure out what your drafting probabilities ought to be.

A hard win condition is a card that you absolutely have to draw in order to win most of your games, or a card that is an essential part of your normative strategy. Alexstrasza in Freeze Mage or Gadgetzan Auctioneer in Miracle Rogue are archetypical examples, but you can arguably make some broad matchup-related calls here as well, e.g., it’s quite difficult to win games against control decks as Hunter without the burst/reach provided by Kill Command. You might even place Savage Roar in this category given how many Druid games are consistently won through some form of SR combo.

A curve-critical card is a card that you must draw by a specific, identifiable point if it is to be useful in your game. Curve-critical cards tend to be a class (like “two-drops” or “board clear”) rather than specific copies (although Gadgetzan Auctioneer in Miracle Rogue falls into this category as well as win condition), and arguably most cards in your average deck are in some sense curve-critical, making this the broadest and most important category. Openers generally, and crucial early plays like Undertaker and Mechwarper are particularly of note in this category.

Lastly, a tech card is a card that swings games for you, but only under conditions particular to a specific metagame or matchup. Ironbeak Owl and Big Game Hunter. Generally speaking, tech cards are exempt from normal considerations of drafting probability since you’ll only run a copy or two total to give you a slight statistical edge over the course of many games. The only time you need to really think hard about curving properly into your tech is when you’re running a significant amount of tech (3+ cards) to hard-counter a particular strategy that’s extremely prevalent in the current meta and you need to keep your curve from bloating in a bad direction (e.g., in a heavily Shaman-dominant meta, as Hunter running double Unleash the Hounds, double Explosive Trap, and Explosive Shot).

In our decklist, we have no tech cards because Zoo is consistent enough that the most tech it’s generally helpful to run is an Ironbeak Owl. We also have no hard win conditions, and so our cards are probably going to fall mostly into curve-critical categories. I’d suggest these categories, broadly, are “something strong to play turn 1,” “something strong to play turn 2,” “something strong to play turn 4,” and “stuff strong enough to help close the game after turn 4.” This is a pretty standard layout for an aggressive deck, with stuff like Abusive Sergeant and Harvest Golem that can flexibly be put down for optimum value at many points in your average game contrasted with cards like Undertaker which lose most of their value unless they’re put down at very specific points.

Let’s look at Doomguard for purposes of illustration. He costs 5, so he’s not going down on turn 4, which classifies him as a closer. His drawback, however, means it’s often a bad idea to play him turn 5 (which means we need to assume we’re covering for turn 5 with either a 3 + 2 or a 4 + 1-drop). We want an aggressive gameplan, so we already know that 1- and 2-drops will be plentiful in our deck. We want to be able to play Doomguard from an empty hand, which means playing a card or two on the same turn, so we conclude 2 + 5 (or 1 + 1 + 5) = turn 7 as the ideal turn to play him.

Holding the assumption from the last article that you can predict about six cards drawn through by turn 1, with two Doomguards in your deck, you have about an 8.5% higher chance to draw into one by turn 7 versus turn 5. If we had misunderstood what type of card Doomguard is and when we’re likely to play it, we’d almost certainly be — mistakenly! — larding our deck with a third 5-drop, perhaps Fel Reaver, to give us a proper curve for drawing by turn 5 (since it raises our odds of drawing into a 5-drop on turn 5 by close to 12%). Instead, with a proper understanding of how we’re planning to curve, we’ll keep only two 5-drops and instead use that slot for a card that gives us a stronger start and more flexibility, like, say, Power Overwhelming.

Calibrating Confidence

Looking at our table, you’ll see that diminishing returns set in pretty quickly. Let’s think about 4-drops for a moment. The chance that you’ll draw a 4-drop by turn 4 (assuming standard mulligan of six draws by turn 1) is 30%. Running four copies radically increases your odds, to 78.16% — almost fifty percentage points. But adding another four copies on top of that, for eight total, raises your draw odds fewer than twenty points, to 96.52%, and adding another four afterward (for 12 total) only raises your odds to 99.25%.

I’m going to suggest that as a general rule, you almost never need much more than an 85% chance to draw into any card at any cost. Here, again, the Rule of Six seems to hold pretty strongly — as you can see from looking at the table, the only probabilistic reason to run more than six cards at a particular slot is if you need to draw into one or more during your opening hand / mulligan. Zoo, for example, depends on having something to play on the first turn. We have six solid turn 1 plays (Flame Imp, Voidwalker, Undertaker, plus four suboptimal plays (Abusive Sergeant, Leper Gnome). Assuming standard mulligan, that gives us about a 70% chance of drawing into an optimal turn 1, with roughly an extra 20% chance of a suboptimal but still solid turn 1, leaving only about 10% of games where we won’t get a strong start — perfect for Zoo’s requirements.

For important-but-not-critical plays, I’m going to suggest roughly a 75% draw chance as consistent without getting greedy. Since we again want an 85%+ chance in Zoo to have something to put down on turns 2 and 4 assuming standard mulligan, this suggests that we should run 8-9 cards at 2 mana cost, and 4-5 at 4 cost. Most pro Zoo players only run Harvest Golem at the 3-drop slot nowadays, but since we’ve already decided Doomguard isn’t a turn 5 play, our table suggests it might be a good idea to run an extra 3-drop just to improve the consistency of a turn 5 3+2 or 3+tap play.

The takeaway here is to be very careful not to draft redundantly. Look at your confidence level and if it’s pretty high, you can likely put more cards in at other, more practical costs (or put in more tech, etc.). Draft to avoid those diminishing returns (e.g., be extremely wary of drafting more than three 8+ cost cards), but also draft to ensure you have high probability to draw the cards you want on curve.

Class Consciousness

As a final note, it’s important to consider your class power when drafting. For example, Warlock can draw a card and play a minion every turn from 3 onward; this lets you leave card draw out of your deck, but it also means you don’t want to run a whole lot of 3-drops since it’s typically a smarter play to put down a 1-drop and tap on turn 3 if you’re ahead.

You’re also running lots of 2-drops due to your aggressive gameplan, so this means that if you’re NOT ahead, you can generally drop a 2-drop and a 1-drop on turn 3 instead of tapping or putting out a 3-drop. On the other hand, we also know that our only 5-drop is Doomguard and we’re not generally playing it on turn 5, meaning a 3-drop + tap is a pretty important play if we don’t have a solid 4-drop in hand.

This encourages us to run that third 3-drop while at the same time recognizing that more would be totally unnecessary given how well-covered our turn 3 already is via 1- and 2-drops. With careful consideration of our hero power, we’ve drafted a leaner, more effective curve.

This is true in differing ways across classes. Mage, Druid, and Rogue have stronger low-cost removal, relatively speaking, because of their hero power. Priest and Warrior have life gain, which lets them bank a little less on early board control. Shaman and Paladin can summon low-end minions without spending cards, and Hunter can focus on drafting for the board because of his reliable damage to the face. The actual different choices you’ll make depend considerably on the specifics of the archetype you’re drafting, but this explains why — for example — Knife Juggler is practically an auto-include in Zoo and aggressive Hunter builds, while not nearly as prevalent in aggressive / tempo Mage.

Conclusion

Whew. That’s a lot of information to take in all at once! Hopefully you’re finishing this article with a much improved understanding of why we draft particular numbers of cards at particular costs, and how to use these probabilities to your own advantage in drafting. Future articles will go into more specifics on drafting for particular styles of play, but you should now have a solid base from which to start analyzing and improving your own deckbuilding. Get out there and show the world how to build a better meta — I’ll see you in-game!