Limiting Chance: Two-Land Hands

Two weeks ago Dan looked at six-land hands and concluded that they’re terrible. This week he looks at a land count that’s much more contentious: two lands.

Two weeks ago I looked at six-land hands and concluded that they’re terrible. This week I’ll look at a land count that’s much more contentious: two lands.

Keeping a two-land hand is sometimes the right decision and sometimes not. It depends a lot on the other cards in your hand and a little bit on a bunch of other things. Because there is no obvious right or wrong answer, I constantly have to decide again and again whether or not to keep two-land hands. Perhaps some hard number crunching can help prepare me for that decision.

If I start with two lands, it’s always really nice to get a land on my first draw. I know how to calculate the chance of that: 15 in 33, or 45.5%.

Lands in Starting Hand Probability of Drawing a Land First Turn
2 45.5%
3 42.4%

The chance is only very slightly higher than normal and still under 50%. I definitely shouldn’t expect to draw a land first turn.

Stuck at Three Mana

It’s fine if I don’t hit a land first turn. What’s really important is that I don’t miss an early land drop. If I get stuck at two mana, that’s a disaster. If I get stuck at three, that’s not so hot either. So here’s a question worth answering: what is the probability that I will be able to play my third land on turn 3?

Not an easy calculation to do with pen and paper, so I’ll use my computer program again. I should mention that I’m doing this calculation for when I’m on the draw. Here are the results.

Lands in Starting Hand Probability of Hitting Third-Turn Land Drop
2 85.0%
3 100.0%

I’ve got a 100% chance to hit my turn 3 land drop with a three-land hand for the obvious reason that I start with three lands. With a two-land hand, my chance is 85%. That’s pretty high. I’m probably not going to miss my third-turn land when I start the game with two lands.

What about my fourth-turn land?

Lands in Starting Hand Probability of Hitting Fourth-Turn Land Drop
2 62.6%
3 90.5%

The chance of hitting my fourth-turn land drop with a three-land hand is over 90%. Not something I have to be too concerned about. With two lands my chance is much lower: only just over 60%. When I keep a two-land hand, I’ll often end up playing four-cost spells at least a turn later than I’d like. That’s good to know when I’m deciding whether to mull. It helps me evaluate the spells in my hand.

How long am I likely to have to wait for that fourth land? I’ve taken the calculation one step further and computed the probability of having fewer than four lands on later turns.

(The probability of not having four lands is opposite the probability of having four lands. So the numbers in the first row in this table are equal to 100% minus those in the previous table.)

Looking at the graph, I have to say that the chances of being stuck at less than four lands don’t go down fast enough to make me comfortable. Even on turn 5 there’s a 23% chance, almost one in four, of not having four lands. The chances only drop below 10% on turn 7! The probability starts below 10% for a three-land hand. I guess there’s a reason that I squirm in my seat when I keep two lands.

Another way to approach the same question is to calculate the average turn I’ll get my fourth land. Here are the results from my program. (I’ve included one- and zero-land hands just for fun.)

Lands in Starting Hand Average Turn of Fourth Land
4 4.0
3 4.2
2 4.9
1 6.1
0 7.6

Obviously a four-land hand always plays its fourth land on turn 4, so the first row makes sense. A three-land hand isn’t too far behind. For a two-land hand, the average turn is almost turn 5. So I should expect to cast four-cost spells about one turn later than normal. That’s a little bit like saying they all cost one extra mana. In fact, that’s probably a good way to think when I’m evaluating the spells in my two-land hand.

Stuck at Five Mana

Ok. Let me do all that again, but now for six lands instead of four. Here is the probability of not having six lands by a given turn.

Not surprisingly, the odds of my sixth land coming later than turn 6 are much higher than my fourth land coming later than turn 4. Even a three-land hand has a 50% chance of not having six mana on turn 6. For a two-land hand, the chance is about 75%, or three of every four games. Even out at turn 9, there is a one in three chance I will not have six mana. Ouch!

Let’s look at the average turn I can expect to get my sixth land.

Lands in Starting Hand Average Turn of Sixth Land
4 6.5
3 7.4
2 8.6
1 10.0
0 11.3

A three-land hand has to wait until around turn 7 or 8 to cast six-cost spells. A two-land hand has to wait until around turn 8 or 9. As before, the two-land hand is about a turn behind.


The danger of keeping a two-land hand is not that you won’t get to three lands. There’s a very good chance you will. You have an 85% chance of getting your third land by turn 3. But don’t let that dominate your perception of a two-land hand. The real danger comes from the fact that you are probably going to hit every mana count after three about a turn later than normal.

Should you keep a two-land hand? It depends on the other cards in the hand. Can those cards support you through the extra time you’ll probably wait for mana counts beyond three? Obviously you are looking for at least one two-drop to eliminate the chance you’re completely dead in the water. You should assume that you won’t be able to cast four cost spells until turn 5. So to avoid a dead turn 4, you’re going to want at least two three-drops. Finally, treat any six casting cost spells with great suspicion. They will not make it into play until around turn 9, potentially well after the game has swung against you.

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source code for the Magic Probability Toolkit