Limiting Chance is a series that uses mathematics and probability to investigate aspects of Limited play. In this article, I take a look at the hands that come after a mulligan.
When I'm looking at my opening hand, I sometimes feel like a kid who didn't study for his exam. I'm trying to work though all these odds in my head, odds that I should really already know.
If I mulligan, what are the odds I'll get a hand that's even worse? What are the odds I'll have to mull again and drop to five?"
Maybe I should calculate those chances once and memorize them instead of trying to work them out fresh each game. In this article, I begin that process.
Hands After A Mulligan
First let's look at the hand I get after a mulligan.
In my first article, I calculated the probability distribution of different land counts in an opening hand. That was useful information. It's even more useful if I also know the distribution of six- and five-card hands. What kind of hand am I likely to see when I mulligan to six? What about when I mulligan to five?
Here's the probability distribution for a seven-card hand, repeated from Limiting Chance: The Opening Hand. This distribution assumes a 40-card deck with seventeen lands.
Three lands is the most likely count, with there also being a sizable chance of two and four lands.
Repeating the same calculation, but now for a six-card hand, gives the following graph.
The chance of drawing four lands has dropped, and a two-land hand is now just as likely as a three-land hand.
Here is the distribution for a five-card hand.
The peak of the graph sits squarely on two lands. There's also a high chance of drawing one or three lands.
I've merged the three graphs into one so it is easier to see how the land counts change as my hand size drops.
The chance of drawing three lands stays relatively steady across all of the hand sizes. The chance of a four-land hand drops significantly as I go to six and then five cards. The chance of a two-land hand grows, while the chance of drawing only one land spikes alarmingly high.
Here is a table of the probabilities, including four-card hands as well.
Average Land Count
The peak of the above graphs goes down by about half a land each time I reduce the card count. That makes sense since about half the cards in my deck are lands. I can verify that by calculating the average land count for each hand.
The average land count is dropping by exactly 0.425 lands per card in hand. That is equal to my deck's ratio of lands to cards, seventeen over 40.
Flooded and Screwed
How much does the drop in hand size affect the odds of getting mana screwed?
I'll group the land counts into the same categories as my first article to see how my chances of getting flooded or screwed change as my hand gets smaller.
- Screwed: two or less lands
- Desirable: between three and five lands
- Flooded: six or more lands
(Whether two lands should count as screwed or not is up for debate. I'll justify that later in the article.)
What can I learn from this graph? The chance of a bad land count in my initial hand is one in three. That's about once a match. When I mulligan to six cards, I have a 50/50 chance of a desirable land count. That's good to know when I'm deciding whether it's worth the risk to mulligan. At five cards, the chance of a good hand is only one in three. That's quite low. To be worth it to drop to five cards, my six-card hand must be very bad. With four cards, the chance of a good hand is terrible. Only one of every five hands will be desirable.
Here is a simplified table to make the odds easier to remember.
A Competitive Start
Knowing the odds of each hand size having the mana I need is good information. It's easy to remember, and it's important when making the decision of whether to mulligan. Yet there's more to the early game than just the opening hand. The cards I draw during those first few turns are critical. With a bad starting hand, there's a very real chance of still drawing the lands or spells I need. Correspondingly, with a great starting hand, I can still flood out. That's what makes the mulligan decision so hard!
When I'm looking at my seven-card hand and determining what to do, I should be comparing the chance that I'll "get there" with this hand against the chance that I'll "get there" if I mulligan.
What Do I Want?
In order to figure out the probability of "getting there," I need to be clear about what "there" is. What do I want?
I want a strong, competitive start to the game. I want to put up a fight. Maybe my opponent will drop a bomb and win. Maybe they will eke out an incremental advantage and overwhelm me, but I want to make them work for it. I want to have the lands I need to play my spells, and I want to have some spells to play.
To bring mathematics to bear on this problem, I need to define mathematically what I mean by a competitive start. I propose this definition:
- I have at least four spells available by turn 4, either already cast or in my hand.
- I'm not stuck at three mana. I play my fourth land on turn 4.
Basically, four spells and four lands on turn 4. Any less and I feel betrayed by my deck.
Now that I have a concrete definition of a competitive start, I can use the Magic Probability Toolkit to calculate the probability of getting there from different starting hands. Again I'll use a seventeen-land deck with 40 cards, and for now I will only look at games where I am on the draw.
Here is the probability that I will have a competitive start for each land count in an initial seven-card hand.
It is my opinion that the numbers in this table are worth committing to memory. Sometimes numbers that come out of a calculation are only useful to understand a trend or to demonstrate a concept. That is not the case here. These probabilities are real and worth tucking away in my brain. When I am looking at a two-land hand, I should be saying to myself: "Ok, this gives me a 60% chance to get there." Like a professional poker player, I should know the numbers, and I should use the numbers.
Here's the same table rounded to the nearest 5% to make it easier to remember. (Don't forget that these probabilities only apply when you are on the draw. In a future article, I will look at being on the play.)
Since I'm claiming this table is so damn important, let's take a closer look at it. With a four-land starting hand, I have a 98% chance of a competitive start. That's awesome! No wonder I love four-land hands. Having three lands is great too, with a 90% chance. Five lands isn't too far behind. It's close enough that it rounds equal to three lands. The chance falls to 60% for two- and six-land hands and then drops very low for everything else.
Do these numbers make sense? Do they match my expectations?
It makes sense that the chance is very low with a zero-land hand. I'll have to rip four straight lands off the top of my deck to have four lands by turn 4. The chance is slightly higher with a seven-land hand because there are more spells than lands in the deck, so drawing four spells in a row is more likely than drawing four lands.
It's also no surprise that the best chances come from three- and four-land hands. To make the grade, they both only need to hit one land or one spell respectively out of four draws. A four-land hand gives the best chance because there are more spells than lands in my deck.
Knowing the numbers for a seven-card hand is great, but they aren't too useful unless I also know what's up with six-card hands. When I'm considering a mulligan, I need to know what I'm getting myself into.
Here are the probabilities of a competitive start for the various six-card hands. The far-right column is rounded to ease memorization.
Let's delve deep into the table. Not surprisingly, it's worse across the board than seven cards. A three-land hand is now the best draw, coming in at a respectable 90%. That's the hand I'm hoping for when I drop to six cards.
Four lands are fine at 85%, but it's no longer the top draw. The fact that the four-land hand only starts with two spells hurts its probability.
The two-land hand has barely moved, staying at 60%. In both six- and seven-card hands, a two-land hand needs to draw two lands to be competitive. It already has all the spells it needs. Thus two lands have the same chance of being competitive whether in a six-card or seven-card starting hand.
A five-land hand has gotten much worse, dropping from 90% to 55%. It simply needs to draw too many spells.
Zero- and six-land hands remain horrible.
When To Mulligan
Here's where things get interesting. I know the probability of getting to a competitive start for each six-card hand. That's what I just calculated. I also know the probability of drawing each six-card hand. I calculated that earlier in the article. Bringing these two pieces of information together, I can calculate the overall chance of a competitive start across all six-card hands. Here's how I do it.
For each land count, I multiply the chance of drawing that hand by the probability of a competitive start with that hand. Then I total the results:
(Some readers may realize that there is a simpler way to do this calculation. Bear with me for now and you will see in a future article why I chose this slightly more complex approach.)
The overall chance of a competitive start from a six-card hand is about 65%. Now I know precisely what I'm getting myself into when I decide to mulligan. When I'm looking at my seven-card hand, I can make an informed analysis of the situation. I can decide with confidence whether the correct decision is to mulligan.
In fact, let me be more concrete about it. Looking at the earlier table of probabilities for seven-card hands, I can compare their chances with the overall chance from a six-card hand. If my current hand has a worse chance of being competitive than the average six-card hand, I should mulligan. If it's better, I should keep.
The numbers say that if I want to maximize my chance of getting to what I've defined as a competitive start, I should keep three-, four-, and five-land hands. I should mulligan everything else.
Of course, I am not saying that this table is the final, incontrovertible statement on mulligan decisions. Rather, it is just the starting point for any decision. There are many other considerations to account for. That's especially true for a two-land hand, as its probability is only slightly lower than that of a six-card hand.
Whether to mulligan to six cards is the most common mulligan decision, so it's the most important to master. But I also need to understand what to do when mulligans take me down to five cards or lower. I also need to know what to do on the play, not just on the draw. Now that I've presented the basic concepts and explained how the math works, I can dive into those problems in future articles.
There is also the question of my definition of a competitive start. Perhaps that definition is not appropriate for certain decks. For example, there is an explosively fast Avacyn Restored red deck for which a different definition might be appropriate. It wants tons of gas and only really needs three lands. If I changed the definition of a competitive start to be more appropriate to that deck, it would be interesting to know how the resulting rules are affected. Are different mulligan guidelines called for when piloting a deck like that?
Perhaps you feel my definition of a competitive start is wrong even for normal decks. What definition would you suggest?
The average land count with a seven-card hand is about three-and-a-half. Every time you mulligan, the average drops by about half a land.
The chance of getting a good land count when you mulligan to six cards is about 50%. The chance of a good land count when you mulligan to five cards is about 35%.
Memorize these chances of a competitive start for the various seven-card hands when on the draw. Compare them to the average chance across all six-card hands, 65%, when judging whether to mulligan.
A strict reading of the numbers indicates that you should keep three-, four-, and five-land hands when on the draw. You should mulligan the rest.