# Magic Math – Mulligans

The Top 8 of Pro Tour Battle for Zendikar featured several hotly-contested mulligan decisions. Mulligans are one of the most complicated parts of Magic, but often the debates on whether to keep or mulligan center around qualitative arguments. In this article, I will explore some of the quantitative aspects of mulligan decisions: I’ll describe the probability calculations behind them and provide some baseline numbers.

But let’s start with the hands from the Pro Tour that motivated this article. In Game 2 of the semifinals, having already mulliganed twice, Kazuyuki Takimura drew the following 5-card hand on the draw:

Takimura reluctantly went down to 4 cards and miraculously won the game.

Several hours later, in Game 4 of the finals, Ryoichi Tamada found the following 6-card hand on the draw:

Tamada kept the hand, didn’t find a land in time, and lost the game.

Did they make the correct decisions to respectively mulligan and keep these hands? The outcome of their games doesn’t tell us much because that’s just one realization. Instead, we should figure out if mulliganing gave them a higher chance to win the game than keeping. To that end, let’s first analyze two simpler situations to get a feel for the probabilities involved.

# Sample Hand #1

Suppose that you’re on the play with your Standard Abzan deck and get the following 7-card opening hand:

Let’s say that your deck contains 18 lands that produce or fetch green mana. In other words, roughly one third of your remaining deck is a good draw and roughly two thirds of your remaining deck is a bad draw.

### Question: What is, approximately, the probability of drawing at least one green source in any of your first two draw steps?

A. Roughly 1/3*1/3=1/9 or 11.1%
B. Roughly (1/3*2/3)+(2/3*1/3)=4/9 or 44.4%
C. Roughly 1–(2/3*2/3)=5/9 or 55.6%
D. Roughly 1/3+1/3=2/3 or 66.7%

C.

Technically, with 53–18=35 bad draws in the deck, the precise answer is 1–(35/53)*(34/52)=56.8%, but we’re not allowed to use calculators in real games of Magic, and you will have received three slow play warnings by the time you have determined that number in your head. Since an approximation is good enough for most mulligan decisions, I’ll happily approximate 18/53 to 1/3, disregarding the effect of a shrinking deck size, and find an answer quickly. These are the types of shortcuts that I use in tournaments.

Okay, but why is the answer C? Well, let me start by explaining why answer D is incorrect. The easiest way to do that is to see what would happen if we would take its logic further: if we would have 4 draw steps to find a green source, then according to answer D’s logic, the probability of finding a green source would be 1/3+1/3+1/3+1/3. This is nonsense because it exceeds 1, and nothing happens with more than 100% certainty.

For those of you who are not all that familiar with probability calculations, it is important to keep in mind that “and” means multiplication and “or” means addition. So the probability that both event X and event Y will happen (if they are independent) is the probability of X times the probability of Y. And the probability that X or Y will happen (if they are exclusive) is the probability of A plus the probability B.

So, answer A represents the probability of drawing exactly two green sources and answer B represents the probability of drawing exactly one green source, either in your first or your second draw step. We weren’t looking for those numbers—we were interested in the probability of drawing at least one green source—and therefore answers A and B are incorrect.

The correct answer is C: 5/9. One way to arrive at that number is by summing answer A and answer B. But it’s often faster to use the complementary probability, that is, find the probability of drawing no green source at all (2/3*2/3) and subtract that from 1. Either way, we are a small favorite to draw the green source in time.

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### Question: You’re playing against an unknown deck. Should you mulligan this hand or keep it?

My Solution

Keep.

We now know that we have approximately a 5/9 probability of drawing into a green source in time and a 4/9 probability of not finding a green source by turn 3. But that’s not enough for determining whether or not we should mulligan. We need to estimate three additional things:

1. The probability that we will win the game if we keep the hand and find a green source by turn 3. This includes the games in which our first two draw steps are Forest and Canopy Vista; or Windswept Heath and Siege Rhino; or Canopy Vista and Den Protector; etc. Some are better than others (especially since Canopy Vista may enter the battlefield tapped) and sometimes our opponent has a great draw as well. But in most scenarios, our Abzan deck will have a close-to-perfect start, and altogether I’d peg the game win probability at around 70%.
2. The probability that we will win the game if we keep the hand and don’t find a green source by turn 3. It’s not zero because our opponent may stumble as well, our first two draw steps may yield Hangarback Walker plus Plains to buy us time, and/or we may find a Forest in our third draw step. But the overall game-win probability for this set of situations is small, and I’d pessimistically peg it at around 10%.
3. The expected probability that we will win the game if we take a mulligan. This should encompass the games in which we keep a 6-card hand as well as the games in which we go further down to 5 or even 4 cards. My estimate, based on data scraped from Magic Online replays mixed with my own experiences, puts this probability at around 40%.

These estimates are up for debate—they are guesses based on experience, not on thorough analysis—but they allow for a structured decision process that focuses our thinking on the relevant questions. With these estimations, the probability that we will win the game if we keep the hand is 5/9*7/10+4/9*1/10=13/30 or 43.3%. Because these are simple fractions, it is possible (with a little practice) to do this in your head in a few seconds. Since 43.4% is more than 40% (the expected game-win probability after a mulligan) I would decide to keep, although it’s close.

If you’ve ever seen me go deep into the think tank after receiving my opening hand, then it’s likely that I went through this entire thought process and made these types of quick calculations. I’ll conclude my solution with two remarks. First, if our hand would have a much worse composition of cards (e.g., no removal spells) then the win probability if we would draw the green source would be much lower to the extent that the hand turns into a mulligan. Second, if I would only have 12-13 green sources in my Constructed deck (or, almost equivalently, 8 sources of a color in a Limited deck) then I would mulligan a hand like this as well.

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# Sample Hand #2

Suppose that you’re on the draw with your Modern deck and get the following 6-card opening hand:

Between a free scry, our draw step, and the Gitaxian Probe, we have 3 draws to find a land in time. Since Storm decks typically play around 18 lands, let’s assume for simplicity that our library consists of one third blue-producing lands and two thirds other cards.

### Question: What is, approximately, the probability of drawing at least one blue-producing land in the top three cards of our library?

A. Roughly 1/3*1/3*1/3 = 1/27 or 3.7%
B. Roughly 3*(1/3*2/3*2/3)=4/9 or 44.4%
C. Roughly 1–(2/3*2/3*2/3)=19/27 or 70.4%
D. Roughly 1/3+1/3+1/3=1 or 100%

C.

No, this is not a trick question. The explanation is exactly the same as before, except that we now get to look at one more card.

All of this is actually a special case of a general rule: if we have B bad cards in our library of L cards and we have D draws to find a good card, then the approximate probability of finding at least one good card is 1–(B/L)^D. This approximation pretends that we draw with replacements and thus becomes inaccurate for large D, in which case you should take into account the continually shrinking deck with a hypergeometric probability, but it works well for most mulligan decisions.

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### Question: You’re playing against an unknown deck, and your opponent kept their opening seven. Should you mulligan this hand or keep it?

My Solution

Keep.

I’ll quickly estimate the three key probabilities once again.

1. I’d guess that the probability of winning the game if we keep the hand and find a land in our top 3 cards is around 60%. A good hand for a combo deck that contains all the key pieces is typically better than a good hand for an Abzan deck, but 6 cards is less than 7, we are on the draw, and we still need to find a second and possibly third land in time. Taking all this into consideration, my estimation of 60% for the game win percentage in this case is a bit lower than for the Abzan deck.
2. I’d guess that the probability of winning the game if we keep the hand and don’t find a land in our top 3 cards is around 5%. Modern is not a very forgiving format.
3. I’d guess that the probability that we will win the game if we take a mulligan to 5 on the draw is around 28%. (It would be several percentage points lower if we were on the play.)

With these estimations, the probability that we will win the game if we keep the hand is 19/27*60%+8/27*5%=43.7%. This is not an easy calculation to do mentally if you want the exact number, but if you replace 19/27 by 70% and replace 5% by 0% then it’s not too difficult to quickly arrive at a number that is close enough. Since 43.7% is more than 28%, I would easily take the risk and keep. In fact, I would probably keep the hand even if I didn’t have the Gitaxian Probe or if I were on the play!

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# Two Useful Tables

The preceding two sample hands were simple in that we only needed one land to get started. But sometimes we’re interested in the probability of drawing two or even three lands, which is much more difficult to estimate on the fly. I’ve never attempted it in a tournament setting.

But it can still be useful to just know a few numbers by heart. Poker is a good analogy. If you ask a professional poker player what their win chances are if they go pre-flop all-in with AK suited vs. 22, then most of them will know the answer. Not because they can mentally sum all the relevant probabilities in a second, but because the situation is common enough to know the probability by heart.

While we’re not always playing with the same 52-card deck in Magic, it can still be useful to know some probabilities for a deck which contains 45% land. That is roughly the ratio you get in a 53-card Constructed deck with 24 lands or a 33-card Limited deck with 15 lands—typical occurrences after drawing your opening hand. For these situations, a binomial approximation (such that deck size doesn’t matter) yields the following.

I bolded three probabilities here that may be useful to learn by heart:

• If you have 2 draws to find at least 1 copy of a certain type of card which makes up 45% of your library, then there’s a 70% chance of success.
• If you have 3 draws to find at least 2 copies of a certain type of card which makes up 45% of your library, then there’s a 43% chance of success.
• If you have 4 draws to find at least 3 copies of a certain type of card which makes up 45% of your library, then there’s a 24% chance of success.

Another useful thing to know by heart are the win percentages you can expect when you mulligan. Although the true numbers depend on the format, your deck, the matchup, whether or not your opponent mulligans, and so on, you can take the ones in the table below as a baseline estimation. They are based on data collected by Rolle’s replay scraping bots in various formats over the last year, on my own experience, and on an attempt to adjust old numbers for the impact of the Vancouver mulligan rule. That last part leads to a bit of guesswork on my part because I haven’t seen data after the new mulligan rule went into effect, but I’d expect these numbers to be reasonable to within a few percentage points.

So, when you decide to mulligan your original seven (and there’s still the possibility that you have to mulligan your 6-card hand as well) then there’s approximately a 40% probability that you’ll win the game.

With all these tools at our disposal, now let’s return to the hands from the Top 8 of the Pro Tour.

# Takimura’s Hand

### Question: You are on the draw. What is, approximately, the probability of having three lands in play on turn 3?

Around 14%.
The original version of this article had 24% as the answer, but that’s wrong because if you’re looking for more than one copy of a certain card, then scrying is not the same as drawing a card. After all, if the top of your deck is land-land-spell-land, then a scry and three draw steps won’t get you there. Instead, we must first find at least one land in two cards (which happens with probability 70%, taken from the table) and then afterwards we must hit 2 lands in 2 draw steps (which happens with probability 20%, taken from the table). The multiplication of these two numbers comes out to 14%.

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### Question: You are playing the Abzan mirror and your opponent kept six cards. Should you mulligan this hand or keep it?

My Solution

Mulligan.

If you have three lands in play by turn 3, then I’d estimate that your expected probability of winning is around 40%. (That may seem low, but you still have one fewer card than your opponent. Moreover, even if your mana situation works out, you’re still in rough shape because the cards in your hand are actually not that good in the mirror: you don’t have a good fighter for Dromoka’s Command and you lack a good defense against a reasonable curve with hard-hitting threats.) If you don’t have three lands in play by turn 3, then I’d estimate that your expected probability of winning is around 5%. This is a bit of a wild guess, but it’s a weighted average of being 15% to win if you find a third land on turn four and a 3% to win otherwise.

Given these estimates, a keep would lead to an expected game win probability of 14%*40%+76%*5%=10%. By contrast, if you take a mulligan, then your probability to win will be a little higher than 15%. (That’s the number from the table, but that’s for a typical game, and in this game your opponent already went down to six cards.)

I should note that a lot hinges on my game win probability estimates, which are merely guesses based on intuition. They are easily the weakest part in the entire logic because different players may have different experiences and arrive at different estimations. But by breaking up the loaded question of whether to mulligan into several sub-questions about the win probability in specific situations, we can take the bull by the horns and foster a more focused discussion, which is what I’m trying to accomplish.

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### Question: You are on the draw. What is, approximately, the probability of having two lands in play by turn 2?

Around 32%.
It’s not 43%, as I initially mistakenly believed—just like with Takimura’s hand, scrying is not the same as drawing a card. Instead, we must first find at least one land in two cards (which happens with probability 70%, taken from the table) and then afterwards we must hit a land in our second draw step (which happens with probability 45%). The multiplication of these two numbers comes out to 31.5%.

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### Question: You are playing against Abzan and your opponent kept their opening seven. Should you mulligan this hand or keep it?

My Solution

Mulligan.

If you find two lands by turn 2, then I’d guess that your probability to win the game is around 50% because, if you draw the right lands, you have a fine hand: a turn-2 Jace that can dig a card deeper to hit the turn-3 Mantis Rider. If you don’t find a second land by turn 2, however, then I’d peg your win probability at only around 5%.

Given these estimates, a keep would lead to an expected game win probability of 32%*50%+68%*5%=19.4%. Comparing that with a 28% to win if you mulligan, which is taken from the table, I conclude that a mulligan gives a higher probability to win.

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# Conclusion

Mulligan decisions remain tricky, but I hope that you’ve found some helpful ideas in this article. And in the end, if you ever wonder whether you have made a mistaken mulligan decision, remember: There are no mistakes, only happy little accidents.