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From the Deck: Mulling Mulligans

I’m an old-school Magic player—I haven’t seriously played since The Dark expansion—so the concept of the mulligan was something I wasn’t very familiar with. I knew how it worked in theory, but I always felt I could use a more in-depth look at it, i.e. when I should be keeping hands vs. when I should be pitching them.

The first thing I want to say is that every deck is different. A “keeper” for one deck might be a “must-pitch” for another, simply due to threshold requirements or resource curve. If your deck has a lot of multi-thresholds then you need to take that into account when mulling.

The percentages here are derived from Stat Trek’s hypergeometric calculator, which is a site that you can use to help determine the percentage chance of hands and draws happening with the deck as you have set it up. For the sake of simplicity, I am going to do all the calculations here based on a 60-card deck with a 24-card resource-base.

The Basics

Now if you came here looking for a rule-of-thumb for mulligans, the best I can do for you is “Keep any hand with 2 to 5 resources”. 0, 1, 6, and 7 shard hands should be instant mulligans, but only on your first, seven-card hand.

2.16% chance of 0 shards
12.1% chance of 1 shard
14.26% chance of 0 or 1 shards
1.25% chance of 6 shards
.08% chance of 7 shards
1.344% chance of 6 or 7 shards
15.6% chance of a 0, 1, 6, or 7 shard hand

With a 15.6% chance of having a must-pitch hand in your opener, you are looking at an 84.4% chance of having a keeper. But even then you should look at that “keeper hand” and do the following analysis of it:

  • Do I have the resources and thresholds to play the cards that are in the hand already?
  • Do I have the thresholds to play the core cards in the deck?
  • What are my chances of drawing those thresholds by the time I get to the turn where I can play the cards in my hand?

If your answer to the first question is “yes” you should probably keep the hand regardless of what cards it actually contains. Remember, just because you have a “bad” hand in terms of the cards you keep, your future draws should be statistically better because the number of useful cards left in the deck is proportionally larger.

For the next question, we move into judgement call territory. If you have multiple key cards at a double threshold (like having Xentoth’s Inquisitor and Extinction) then getting that threshold is key to your success. In this case, having no Blood threshold is probably a signal to pitch the hand and try your luck at six cards instead.

With the last point, we can head back to the Hypergeometric calculator and take a look. Let’s assume we have XI, Extinction, and Darkspire Punishers in the deck (all BB threshold), and we have 1 Blood and 2 other resources in our opening hand.

With 53 cards left in the deck, and 11 of those being Blood Resources, over the next 4 turns we have a 61.78% chance of drawing that second Blood. If we are playing with dual shards that goes up to 68.79%, because there are now 13 “successes” in the population. (Protip: Play with Dual Shards, the math works in your favor).

The Six-Card Hand

If you pitch your hand, you are looking at an entirely new set of values for your 6-card hand. Your brain is probably sticking to the mindset of the 7 card hand “rule of thumb” but we have to realize that getting a keeper hand just got harder, since there are less cards.

3.89% chance of 0 shards
18.07% chance of 1 shard
21.96% chance of 0 or 1 shards
3.05% chance of 5 shards
.2% chance of 6 shards
3.25% chance of 5 or 6 shards
25.21% chance of a 0, 1, 5, or 6 shard hand

Wow. We went from a 84.4% chance at a keeper to a 74.79% chance at a keeper. Not only that, but in any keeper hand we are one playable card down then we would have had. And the percentages get even worse if we mulligan down to 5.

The Five-Card Hand

“Never mull below 5” is something you hear said over and over. And it’s true, your chances of winning get smaller and smaller every time your starting hand gets smaller and smaller. Yes, you can win with a 3-card starting hand, but it’s going to be an uphill battle that begins topdecking very early on.

If you pitch a six-card hand, you should probably keep the five-card hand regardless of what it contains. Hopefully you get 1 shard out of it, with 2 being ideal.

25.88% chance of 1 shard
36.08% chance of 2 shards
68.87% chance of 1 or 2 shards

So the odds are pretty good that a 5-card hand is a keeper. With a 2-shard hand you have a 40% chance of drawing a shard on your first draw, and a 41.81% chance if your hand had 1 shard to start. Every non-resource you draw increases your chance of drawing a resource from that point forward (as there is a greater ratio of resources to non-resources in the deck).

Even if your 5-card starting hand had 0 resources in it, you can still keep the hand. You have a 68.68% chance of drawing 1 or 2 resources in your first two draws of the game. If you were on the play you will definitely be behind the curve.

Using the percentages we worked up to this point, the chance you have of having an unplayable hand in any game is 0.53% (Meaning a 7 and 6 card hand without 2 to 5/4 resources and a 0 or 5 resource 5 card hand). Just chalk it up to RNG-sus and do the best you can in these games you’ll find yourself in 1 in every 200 times.

When Should You Keep a 1-Shard Hand?

I am glad you asked. The answer you are probably conditioned to think is “never”, but that’s not true. In addition to any five-card opening hand, there are some six- and seven-card opening hands with one shard that can be considered keepers if the cards in them can either fetch resources (like Immortal Tears), add threshold (like Royal Herald or Adaptable Infusion Device), or your hand is completely or near-completely playable with a single extra shard draw. You can be even more generous in all your estimations if you are on the draw. Remember, having 1 shard in your first 7 cards leaves 23 shards in the remaining 53 cards in your deck. Even if you flood at this point, you kept 6 playable (hopefully super-playable) cards to crush your opponent with.

Other Factors

Knowing the counts and ratios of cards in your deck is key to making good mulligan decisions. If dealt a “keeper” hand of 3 resources and 4 other cards, you may still want to mulligan if the 4 other cards are expensive and/or actions that aren’t useful in the early game. Keeping hands with 2 Sapphire Shards in them because a lot of your key cards have a double Sapphire threshold can be a smart move as well. You’re preparing for a future where you draw into those cards and you don’t want to risk not being able to play them due to threshold costs.

Also, knowing how your opponent’s deck is playing is a factor as well. If they are playing an aggressive deck, then you may want to mulligan for early answers/blockers. You can sometimes tell what style of deck they are playing based on their Champion, so you can make an educated guess—even in game 1—whether to fish for early-game cards with a mulligan.

In these cases, I really wouldn’t mulligan below 6 unless the 6 card hand is unplayable. You don’t want to risk being behind in health, troops, AND hand size because you mulliganed too aggressively.

Using the Hypergeometric Calculator

This final section instructs you on using the Stat Trek site to calculate your own percentages. This is handy if you want to see what shard breakdowns you should do for your decks, or if you want to calculate odds for drawing a single card out of your 60 or 40 card deck.

Population Size

This is the size of the deck. 40 for limited, 60 for constructed, 75 for Jank Bot in limited, and 150 for Jank Bot in constructed. If you believe that you can get away with larger decks, plug the new deck size number in and see how the percentages shake out. Once you see how your odds get worse for every card over the minimum that you include, you’ll (hopefully) gravitate towards running minimum deck size from now on.

Number of successes in population

This is the total number of cards you are determining the odds of drawing. For shards, this is the number of either a particular threshold, or the total number of shards in the deck. In a two-shard deck that holds 10 of one shard, 10 of the other, and 4 of the dual shards, you would have 14 successes for shard 1, 14 successes for shard 2 and 24 success for any resource. If you are determining the odds of pulling a certain card by a specific turn, then you just put the number of that card that’s in the deck here.

Sample size

This is the number of cards drawn. If you want to see “by turn X” then add X to the sample size of your starting hand (if you were on the play). For example the odds of drawing at least one Guardian Angel by turn 6 in a 60 card deck, when on the play, is 63.42%, assuming you have 4 in your deck. (Pop size 60, Successes 4, Sample 13).

Number of successes in sample (x)

This is how many of the card you want to have in that sample size. The calculator will give you all the different variations on X that you could want (equal to, greater than, less than, and combinations of these). This is where you would put the number of shards you want to have if you are using the calculator for shards, but for individual cards you are just looking at putting a “1” here, and looking at the “X > or = 1” line in the results.

So there you have it. If you have any questions remaining about mulligans I’m not surprised. I’ve only covered the basics and tread very lightly on the nuances. Every hand should be looked at differently, so even the guidelines presented here can be tossed to the wind if you know your deck or want to take a risk.

Since HEX is a game of chance and the card draws are completely random, there will be times where your deck ‘defies the odds’ and I want to assure you that’s a good thing for the game as a whole, even if it means you lose because of it. Feel free to ask further questions about mulligans in the comments and if there’s enough interest I can do a follow-up article answering them in better detail.

Matt Miller, known as DeckOfManyThings in-game, is a professional MMO designer by day, and a Hex Kickstarter backer by night. He's been playing TCG's since being taught by Peter Adkinson how to play Magic: the Gathering with Alpha cards, and is now devoting a lion's share of his free time to playing and improving his skills at Hex. Find him at @ManyThingsDeck on twitter or /u/ManyThingsDeck on reddit.

2 Comments on From the Deck: Mulling Mulligans

  1. The Dark was when I started playing MTG. You missed out on some fun, competitive times, Matt. 😀

    Great article. It’s nice to see the math behind our mulligan decisions.

  2. Thanks for the article

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