HearthMath: The Numbers Behind the Mulligan

Today I want to delve a little bit deeper than usual into the mathematical engines of probability - the science of Mulligans.

Introduction

Hey, guys. It’s RaFive. Today I want to delve a little bit deeper than usual into the mathematical engines of probability that run Hearthstone behind the scenes. Specifically, I want to discuss how probability affects drafting for the early turns of the game, particularly in context of the mulligan.

Anyone who’s watched a professional Hearthstone player in a stream or tournament will recall seeing an occasional mouse-over of the deck and some thinking as to the probability of drawing into particular cards. It’s realtively easy to say “I’ve got 20 cards left in my deck, so the probability I’ll draw Ragnaros the Firelord next turn is 5%,” but it takes a little more nuance to plan over the course of several turns, or evaluate your chances when there’s more than one card that might help you out. I aim to go through the basics of developing this skill.

Why learn at all? Well, for players who just want to get out there and play, netdecking works pretty well and will get you decently high up the ranks if you’re a competent player. However, if you understand the math that underlies the decision to put a certain number of cards at each mana cost for control, tempo, rush, etc., then you can begin to build your own decks, having confidence that you can draw into the right cards at the right time. Even if you only have an interest in netdecking your way to Legend, a better understanding of the math behind the cards you’re playing will help you tech in a way that lets you draw your specific advantages more consistently.

Most of this article is about theory, but for occasional reference I’ve included what I consider to be one of the strongest midrange decks currently in the metagame, Bryan Kibler’s Mech-Mage. The curve Bryan has drafted is impeccable, and makes a great case study in drafting so tight that it makes it unnecessary to pack card draw like Arcane Intellect. Onward to the numbers!

The Basic Math

Fortunately for folks like me with poli-sci backgrounds, Hearthstone’s math of the draw is mostly stuff that a reasonably bright college-educated adult can do in his head.

You have 30 cards in your deck, which means you have a 1/30 or 3.33% chance to draw any particular card (1/15 or 6.67% if you run two copies). You can never assume you’re going second, so when you draft, assume you’ll draw three cards on your first turn, which you can mulligan back. Then you draw one card and your first turn begins. That’s up to 7 cards you can draw through for your first turn. Each additional turn adds at least one card you can draw.

Now, here I’m going to cheat the system just a little. Technically speaking, you start with 30 cards, so your odds are 1/30 (3.33%) to draw any particular card. After that, you have 29 cards left, so your odds are 1/29 (3.45%) to draw any particular card. After that, your odds are 1/28, and so on for each additional card that comes out of your deck.

In practice, however, for the first few draws, doing it one at a time like that actually gives you very little gain in accuracy compared with just dividing by 30. For example, the odds of drawing any particular card over six draws tally up to 18.33% if you do it one at a time (1/30, 1/29, etc.), but 6/30 odds is 18%, meaning the shorthand math is only off by an insignificant amount. Being able to estimate your probabilities more quickly is crucial when you’re in-game and have 90 seconds to plan all your moves out, so if you’re looking to become an advanced Hearthstone player, I strongly recommend you get in the habit of doing this shorthand math whenever you’re in-game and thinking about a particular card you want to draw into. (This also has the added benefit of teaching you to count your cards and maintain awareness of what’s left in your deck, which is another crucial skill to develop in advanced play.)

Now that we’ve gone through the sorts of calculation we’ll be doing, let’s apply this knowledge practically to inform how we build our decks.

The Rule of Six

I’m going to recommend that when you build a deck, you assume there will probably be one card in your opening hand that you think is worth keeping because of a particular matchup or a particular opening strategy (like, say, Undertaker). This means that you’ll draw five cards on your mulligan and one on your opening turn, for a 6/30 or 18% (really, 1/5 or 20%, but remember, we’re doing shorthand math) chance to draw into any particular card. This leads us to the first manifestation of what I’ll call the Rule of Six: Draft assuming you can draw up to six cards by turn 1 looking for the card you want.

In your mulligan, you’re looking for something to play generally on your first 1-4 turns depending on deck. (Only in the rarest and most specific circumstances will you ever want to hold a card costing 5+ mana, and unless you’re playing a pretty greedy deck, you generally won’t be holding a lot of 4-cost cards, either.) The most important card in your opening hand, as a general rule, is the card you put down on your second turn. Your play here typically determines who has initiative over the next several turns, and oftentimes can snowball you into outright victory. Therefore, it’s absolutely crucial in most decks that you have a strong turn 2 play available.

What we’re looking for, then, is the ability to mulligan into one of our 2-drops almost every time we play, without fattening our curve there so much that it prejudices our ability to fit higher- (or lower-) end cards into a deck. This leads us to the second manifestation of the Rule of Six: If you’re drawing six cards on average, you can pretty reliably mulligan into a card of a particular cost if you’re running at least six cards at that cost in the deck. You have to do hypergeometric probability in order to derive these numbers, so I’ll do the math for you:

-If you put ONE copy of a card at a particular cost into your deck, you have a 20% chance of being able to mulligan into it.

-Two copies raises your odds to about 35%.

-Three raises your odds to about 50%.

-Four raises your odds to about 60%.

-Five raises your odds to about 70%.

-Six raises your odds to a little over 75%, which means three out of four games, you’ll be able to mulligan into one card at that cost.

That last is about the level of consistency we’re looking for for tempo/midrange drafts, and it’s definitely a floor for drafting aggression, as well (I’ll probably put together another article at some point and assemble full tables so y’all can examine your probabilities comprehensively). Things are a little trickier for control, but the Rule of Six generally applies; just take a look at the Kinguin tourney decklists, for example, and count copies of of “this is a good card in your opening hand” for each list. Most pros won’t go below that magic number for fear of losing draw consistency at one of the most determinative stages of a match.

To recap, then, the Rule of Six is bipartate: (1) Assume you can draw up to six cards before your first turn, and (2) consequently, put in at least six cards if you want to be able to reliably draw into one of those six on turn 1.

Cost vs. Timing

It’s crucial to remember that while mana curve is extremely important in Hearthstone, especially for the first few turns, many cards are not meant to be played at cost. For example, Ironbeak Owl costs 2 but isn’t really a 2-drop since he’ll typically come out to silence Sylvanas Windrunner on turn 8, or some such.

Consider the Bryan Kibler list for a moment. It actually carries eight cards at the 2-mana slot — six minions and two of Frostbolt. However, while you’ll occasionally use Frostbolt on turn 2 to remove a snowballing Undertaker or some such, you’ll frequently be holding it in hand as extra reach or a trigger for Archmage Antonidas (or the opponent won’t play anything worth the card by turn 2). Since you won’t always be running it out on curve, Frostbolt is, therefore, NOT actually a turn 2 play and should be partially or wholly excluded from our Rule of Six considerations in drafting, hence the six 2-cost minions, all of which you’re happy to put out on turn 2 in almost every case.

While it’s important to consider whether a card is likely to be held, of course, it’s also important to account for the earliest turn on which you’re likely to play it. Let’s go back to Frostbolt — there are quite a few instances where you WILL play it on turn 2, which makes it partially a 2-drop but not wholly so. This is the way to think about drafting cards in Druid, for example, where Innervate and Wild Growth are both cards which basically let you play minions like Chillwind Yeti and Keeper of the Grove as 2-3-cost cards. Instead of looking at a Ramp Druid list like this one and thinking it’s horrible on mulligan because it only runs four 2-cost cards, look instead at the timing of when cards can come down, and you’ll see that it actually still fits into the consistency demanded by the Rule of Six because of flexible cards like Innervate that expand your possibilities for what’s “on curve” on any given turn.

Scaling

While for simplicity’s sake I’m concentrating on the first couple turns of the game, all the math here can be expanded for insight into drafting at higher mana costs. For example, with the Kibler mech-mage, how’s our high end? What are our odds of drawing into Archmage Antonidas by turn 7? Do we need another big drop?

Applying the Rule of Six as before, by turn 7 we’ll have drawn at least 13 cards. However, Antonidas isn’t really a turn 7 play because of his ability — let’s assume turn 9 for most games to permit our calculations for a combo off Frostbolt without relying on Spare Parts (since you might always get Time Rewinder and be unable to pay 8 mana for an extra copy of Fireball in your deck). That means we have exactly a 50% chance of drawing into Antonidas on our optimal timing; not great given our deck has no card draw, so we need an alternate win condition. Enter Fel Reaver — with two Reavers plus Antonidas and 12 cards in which to draw one of the three, our odds end up much the same as drafting earlier cards at similar ratios, giving us, again, a little over a 75% chance to draw into one of the three “on curve.” That’s consistent enough to round out our deck (and here, since you always want to draft for less optimal/lucky situations than average, we’re not even taking into consideration the potential cost reduction of Reaver via Mechwarper).

You can scale down with this, as well. Having a turn 1 play is nice in a midrange deck, but you want to keep it to an absolute minimum to make room for the bigger stuff. It’s not quite as critical as your 2-cost slot and plays much worse in the later game than more expensive stuff, so you can afford to run fewer total copies. Two copies each of Cogmaster and Clockwork Gnome synergize nicely with the deck on most turns and don’t overload your curve while still giving you better than even odds of being able to mulligan into them.

If this is getting a bit too heavy, though, fret not — here’s a cheat sheet. Draft four plays at 1 mana, six at 2, four to six each at 3-4, and at least two at 6+, with the rest as your staples and utility cards. If you put in solid cards with good synergies, you’re likely to have a solid, playable deck that gives you an excellent curve for strong, on-tempo plays. This is also, incidentally, not a bad template for a beginner’s Arena.

Conclusion

Your takeaway from this article, if nothing else, should be: draft at least six plays for turn two if you want a reliable mulligan. Past that, however, we’ve delved into the math behind the draw and provided some shortcuts and practical applications for using draw probabilities to design better decks. Now get out there, apply these lessons, and build a better metagame!