Evaluating opening hands is a hybrid of science (math) and art (experience). I’m going to walk through both, but I tried to keep the math-ing to a minimum and just included summaries. If your eyes start glazing over, try to pull out the conclusions as they are still useful.
Mulligan decisions and line analysis involve a lot of probability to calculate optimal behaviors. The specific mathematics you want are hypergeometric distributions (a.k.a. fancy math). I’m just going to put in the percentages for the outcomes, but I’m happy to explain how I calculated them in the comments if there are questions.
Here are my assumptions when evaluating hands:
- Turn-2 Titan is basically a win (our pseudo-kill from the last article).
- Turn-3 Titan makes you a favorite in the game and is a desirable outcome.
- Turn-4 Titan with an Amulet makes you a favorite on play or draw.
- Turn-4 Titan without an Amulet makes you a small favorite vs. the field on the play, but a moderate underdog on the draw.
These aren’t set in stone and they also change post-board, when we can actually interact with our opponents’ game plans, but they provide a good guideline for thinking about mulligans.
The Optimal Hand
Let’s start by laying out your target hand. This isn’t the best possible hand (you can kill on turn 1, after all), but it is a hand that gives you a turn-2 attacking Titan.
- Non-bounceland (16-17 copies) or Simian Spirit Guide (1 copy)
- Bounceland (10 copies)
- Amulet of Vigor (4 copies)
- Summer Bloom (4 copies)
- Primeval Titan/Summoner’s Pact (8 copies)
There you have it. I’ll going to refer to this throughout the article as the optimal hand. The turn-2 pseudo-kill only requires 5 cards, and most of them are fairly redundant. Your number of non-bouncelands that count varies with play vs. draw, as Vesuva usually counts on the draw but not on the play (since you can copy their land). Simian Spirit Guide can play the role of the non-bounceland (giving us a turn 1 pseudo-kill), but doing so leaves you with pretty limited follow up options.
Also note that Ancient Stirrings helps find your most redundant pieces (the lands) as well as your Amulets, while your blue manipulation spells can dig for whatever you need.
The first thing you should do when you look at your opening hand is to see how far you are from optimal. This may surprise you, but the least important piece for you to have is actually the Amulet itself. If you don’t have Summer Bloom (or sometimes Azusa), Amulet doesn’t do very much, so it is the least individually powerful of the combo pieces. On the other hand, Summer Bloom can still be fantastic even without an Amulet to power it.
Today I’m going to look at some hands that are missing the Amulet. These hands can still be very good and it is important to think through how they play out. It is also important to remember that you will always have at least a 7.5% chance to just draw an Amulet at any point, assuming 4 are still left in your deck and your deck is random. For every card you put to the bottom, that percentage goes up. You can frequently spike an Amulet to turn an okay hand into an optimal one.
These draws will never be truly explosive, as your Titans won’t be able to give themselves haste, but you can still easily just cast a turn-3 Titan and attack with it on turn 4. Let’s look at some examples and figure out what makes for a successful non-Amulet hand.
Sample Hand A
This hand gives you a turn-3 Primeval Titan with the following line:
- Turn 1: Tolaria West
- Turn 2: Gemstone Mine, Summer Bloom, Simic Growth Chamber (bounce Tolaria West), Gruul Turf (bounce Gemstone Mine), Tolaria West
- Turn 3: Gemstone Mine, Primeval Titan (likely fetching bounceland and a value land, bouncing Tolaria West)
Based on my assumptions at the start of the article, this is going to beat most Modern decks. I’m going to call this the fair hand (missing only Amulet). It isn’t optimal, but it is still more powerful than most fair decks and you are happy to get here.
Let’s make your hand worse.
Sample Hand B
This hand is clearly worse than your fair hand, as you aren’t guaranteed to play your Titan on turn 3. A quick way to evaluate this hand is to add up the total mana your lands can produce. After you check to see how close you are to optimal, you should look to see how close you are to fair.
Here you can only make 5 mana, so you know you need to draw one more land. If you draw the land on turn 2, it can be any land—if you draw it on turn 3, it needs to be an untapped land.
Assuming you’re on the play, you are 43% to draw a land on turn 2 and 17% to draw an untapped land on turn 3, if you miss on turn 2. Given your 8% chance to draw Amulet on turn 2 (yay for optimal hands), plus all of the library manipulation spells you can draw, your odds of getting a turn-3 Titan here are actually very good. For example, if you draw Sleight of Hand on turn 2, your sequencing is:
- Turn 1: Khalni Garden
- Turn 2: Gemstone Mine, Summer Bloom, Simic Growth Chamber (return Mine), Gemstone Mine, Sleight of Hand
If Sleight of Hand hits a bounceland, you play that and pick up your Gemstone Mine. If it hits an untapped land, you just play your Tolaria West and say go. If it bricks completely, you can complain about running bad and hope to draw an untapped land on turn 3 (19% of the time it works every time).
So even missing a land and an Amulet from optimal, your future is still very bright.
Let’s walk through one more example to evaluate how the library manipulation cards can impact your analysis:
Sample Hand C
What do you do here?
This hand is kind of like Sample Hand B, but there’s only have 4 mana in your hand so you need to find yourself 2 more lands to have your turn-3 Titan plan. You still have 4 of the 5 pieces for your optimal hand, missing only an Amulet, so you are at most an 8% shot away from just getting there. That is always a nice rebuy to have.
How do you evaluate it?
You need to figure out how much work Ancient Stirrings actually does for you. Hint: when you are this close to optimal and fair, Stirrings is a beast. The further you are away from those hands, the less helpful it becomes.
Let’s assume you draw blanks for the moment.
You can play turn-1 Tolaria West and then on turn 2 play Gemstone Mine, cast Summer Bloom, and then play Simic Growth Chamber (bouncing Mine), replay the Mine and cast Ancient Stirrings. You will find a bounceland from the Stirrings 63% of the time, giving you a fair hand. If you fail to find a bounceland, you’re way over 90% to find a non-bounceland for the turn, leaving you just 1 mana short of casting a Titan (Sample Hand B).
Luckily, you don’t actually have to ignore your draw steps.
After you play your turn-1 Tolaria West, you will draw an Amulet on turn two 7.5% of the time, moving straight to an optimal hand. If you don’t, you have roughly 17% to draw a bounceland and go straight to a fair hand. That adds a lot of percentage to your already good-looking odds.
So 7.5% of the time, you end up with an optimal hand, and roughly 70% of the time you end up with a fair hand.
That Ancient Stirrings will find you a copy of Amulet 34% of the time. If you miss, your turn-2 draw step will be an Amulet 8.3% of the time, meaning you will hit an optimal hand just over 42% of the time with this start.
How bad is it if you miss?
It is a lot harder to get to a fair hand with this line. If you want to play a turn-2 Summer Bloom, you need to find an untapped land and you have 6 chances to do so (5 cards from Ancient Stirrings, 1 card from your draw step). You know these 6 cards don’t include an Amulet (or you would have already moved to your optimal hand), so you have about a 68% chance to hit an untapped land and no Amulet, which would let you cast turn-2 Summer Bloom and get a guaranteed 5 lands in play. This gets you to Sample Hand B. To actually hit a fair hand, you need to then also draw a land for the turn, on turn 2 (or get a tapped land from Stirrings and draw an untapped land), which will happen about 50% of the time.
So what is the total breakdown for this line?
- 42% of the time, you get an optimal hand (yay).
- 37% of the time, you’ll hit a fair hand.
- 21% of the time, you won’t be able to play a turn-3 Titan (but will usually be able to play a turn-4 Titan).
Sign me up! The takeaway here is that you should aggressively try and hit an optimal hand in spots like this, rather than maximizing your chances for a turn-3 Titan. Even if you fail, you’ll still fall back into the turn-3 Titan a pretty high percentage of the time.
So it turns out that both Sample Hand B and Sample Hand C were quite good. What can we learn from that?
This deck is not a one-trick pony. You will have a ton of great hands that don’t include Amulet in them. First, Summer Bloom plus lands gives you a great fall-back plan of a turn-3 Titan and you can always draw an Amulet to make your hand unfair.
Additional lands are largely interchangeable with Ancient Stirrings, as long as you can sequence your turns to cast it on turn 1 or 2 without disrupting your curve.
We’ll keep looking at starting hands next time as we discuss what happens when the other pieces (or multiple pieces) are missing.