It’s Round 15 of a Modern Open. You’re 11-3 and in need of one more win to lock up that coveted top 8 berth. You’re playing your trusty Izzet Phoenix deck and entering game 3 against Titan Shift.

The good news is you’re on the play, so a solid hand and you’re in good shape to race in a matchup that has minimal interaction. You fan open the following seven card hand:

Arclight Phoenix

Arclight Phoenix

Faithless Looting

Sleight of Hand

Thought Scour

Lightning Bolt

Polluted Delta

That’s pretty close to the dream seven, right? You have the explosive double Phoenix draw with the enabler in Faithless Looting, and with the number of cantrips in hand it shouldn’t be difficult to trigger those Phoenixes by turn three, providing a clock that plenty of Titan Shift draws can’t keep up with.

And yet, I begrudgingly sent this hand back for another look at six, much to the surprise of Gerry Thompson and Bryan Gottlieb, who were commentating the game in question last weekend in Louisville. You don’t often see commentators so openly question a play on camera, and while I have the utmost respect for Gerry and Bryan as players, I felt compelled to offer the reasoning behind my decision.

For reference, here is how I sideboarded for that game:

Out:

Gut Shot

Gut Shot

Pyromancer Ascension

Pyromancer Ascension

Finale of Promise

Thought Scour

Thought Scour

In:

Blood Moon

Blood Moon

Alpine Moon

Beacon Bolt

Spell Pierce

Spell Pierce

Abrade

As I noted above, the Izzet Phoenix vs Titan Shift matchup is mostly a race, with both decks containing little powerful interaction for the other, and both decks being more than capable of ending the game by turn five. Should this hand come together and trigger the pair of Arclight Phoenixes on turn three, that represents eighteen damage by turn five, where the Lightning Bolt will be lethal if my opponent’s mana base hasn’t already added the necessary two points of damage.

But this hand, while likely to play out that way, is not guaranteed to do so. A missed second land drop spells disaster because there are only four free spells in the deck. Moreover, since we’re forced to spend our cantrips aggressively to ensure we make a second and third land drop, there’s no guarantee that we have three spells left over to cast on the third turn, a concern that is exacerbated by my siding out a pair of Thought Scours to help insulate me from Damping Sphere.

Any stumble on the way to turn three Phoenix triggers and we’re not killing before turn five. Without any disruption that’s a death sentence against the redundancy-fueled efficiency of Scapeshift, so even in the base case this hand is a risk. I’m no stranger to risky keeps in Modern, having kept a zero land, six card hand in the finals of the Baltimore Open last December, but there were several reasons I felt the risk wasn’t warranted in this case.

First is that my deck has the better disruption in the post-sideboard games. Blood Moon and Alpine Moon turn Titan Shift into Colossal Dreadmaw tribal, which, despite its apparent popularity among WotC R&D employees, is far from Modern playable. For this hand those cards are almost a liability, since they inhibit the Arclight Phoenix plan that this hand is mostly all-in on. In an ideal case the Alpine Moon will show up on turn four to prevent Scapeshift from racing the birds, but that’s some serious wishful thinking.

Second is that in this race, I’m going to put myself to seventeen life in order to find Steam Vents off the Polluted Delta, so a seven land Scapeshift dealing eighteen damage will be lethal. That makes a turn four kill significantly more likely, and my hand has little recourse against such a draw from my opponent.

Lastly, and most important in my eyes, a hand that is all-in on Arclight Phoenix is particularly vulnerable to the common sideboard cards in Titan Shift. I already saw Damping Sphere in game 2, and I was also expecting some number of Relic of Progenituses. Either of those cards, cast on my opponent’s second turn, will completely shut my hand down.

There’s some reason to believe that line of thinking is overly cautious, since it’s likely that a ramp spell is a preferable turn two play to either of those. In fact my opponent cast Farseek over Damping Sphere on turn two of the previous game so it’s quite possible that my Phoenixes land safely while the hate card sits awkwardly in my opponent’s hand.

However, with this specific hand, the best turn one play on my end is to lead on Faithless Looting and have access to all of the top three cards of my library before committing a mana on my second turn. That’s the line that offers the highest chance of finding both a second land and a Manamorphose to trigger the Phoenixes on turn two, which is the best possible scenario, and one that a hand as all in as this one has to play towards. Showing the Phoenixes on turn one means my gameplan is face up, and any hate card that stops my plan is going to be cast as quickly as possible.

With no other apparent avenues to victory, this hand has to play for speed and cross its fingers that no Damping Sphere or graveyard hate appears. Trying to play around those cards by not showing the nature of our hand until the last moment slows us down and may not even affect our opponent’s play, especially if they have a Search for Tomorrow on turn one, letting them accelerate while casting a sideboard card on turn two.

Of course, there is significant potential in this hand. Find a Manamorphose and a second land and you have two Arclight Phoenixes attacking on turn two, with a Lightning Bolt to ensure the game ends on turn four. Anger of the Gods is about the only card to be scared of there, and its impact in the matchup is narrow enough that I wouldn’t expect them to be in my opponent’s sideboard configuration, even if they are in the maindeck.

Bryan astutely pointed out that early pressure was important from my side of the matchup, and the lack of it was critical to my loss in game two, so this potential is an important variable, and certainly the strongest motivating factor to keep the hand.

I Did The Math

Now that we have a broad view of the factors in play, let’s make some simplifying assumptions and crunch the numbers.

Calculating the possibility of returning the Phoenixes on turn two is straightforward. Leading on Faithless Looting let’s us look at three cards before our second land drop, and with no Gut Shots or other Phyrexian Mana spells in the deck, we need at least one Manamorphose and at least one land to assemble “the nuts.”

This happens in three distinct configurations:

1 Manamorphose, 1 Land, 1 Other

2 Manamorphose, 1 Land

1 Manamorphose, 2 Land

Since our first land is Polluted Delta that will find a Steam Vents, the remaining 52 cards in our library are the following:

4 Manamorphose

16 Land

32 Other

Given these numbers, there are 4*16*32 = 2048 three card combinations of the first configuration, *16 = 6*16 = 96 three card combinations of the second configuration, and 4*= 4*120=480 three card combinations of the third configuration, where  is the binomial coefficient given by the formula .

From here, calculating the probability of finding one of those combinations is a matter of summing the total and dividing by the total number of possibilities, which is given by

This calculation yields a result of or about one in eight.

 

How likely are we to return the Phoenixes on turn three? That calculation is much less straightforward, but let’s be generous and say that it’s 75% or six out of eight, leaving a fail rate of about one in eight.

I’d say we’re a huge favorite to win in the first case and a huge underdog in the last. Deciding that for the middle case of turn three Phoenixes depends on if our opponent has either a hate card or a turn four kill to win the race.

At the time, I assumed my opponent had three or four copies combined of Damping Sphere and Relic of Progenitus. Looking at his list the number is three, (I’d be surprised if he brought in Grafdigger’s Cage as well) but let’s consider both three and four.

Disregarding the deck thinning of a turn one fetch land and assuming my opponent keeps his seven card hand, he will see nine cards by his second turn. The chance of seeing at least one copy of a four-of in the first nine cards is while for a 3-of the chance is  So we’re looking at about a 40-50% chance of our opponent foiling our one plan of attack--not great.

When you add in the potential for a turn four kill to win the race, that thurn three double-phoenix draw doesn’t look as impressive as it seems. In fact you have to be up towards that 75% figure in order to be about even to win the game, assuming you win about as often with the turn two double-phoenix draw as you lose with a fail rate game. Any less than that and you start to lose percentage to the fail rate games happening more often than the turn two double-phoenix games.

Of course the decision to mulligan this hand also requires us to estimate our chances of winning on a random six card hand. Is Titan Shift a favorable matchup? That’s another very difficult question, and one that’s impossible to answer with a single number, but given how few answers they have for Thing in the Ice and the power of Blood Moon, I liked my chances. Had this been game one without the threat of Damping Sphere and/or Relic of Progenitus (albeit an increased risk of Anger of the Gods) I would’ve kept. That’s how close the decision was.

Now did I calculate the above percentages at the table to arrive at that conclusion? Of course not. I’m not noted mathemagician Arthur Benjamin. But I do know that you’re about 40% to have a given four-of in an opening seven card hand, and I figured that the turn two double-phoenix case was fairly unlikely, while turn three case was probable. These are approximately the numbers that were floating in my head when I made the decision to mulligan.

There are plenty of other minor variables at play here. I could find some lands and a Blood Moon or my opponent could stumble and give me more time to assemble the Phoenixes and race, but these are all much smaller than the variables I considered above.

Sideboarding, like almost every decision in Magic, isn’t an exact science. I find it helpful to perform calculations like the ones above in hindsight because it builds my intuition for later decisions, but there isn’t one final number that will tell us whether that hand is a keep or a mulligan. Maybe one day the machines will take over competitive Magic and enlighten us all, but for now this is a human domain and we’ll have to settle for thoughtful, fully-considered judgement calls.

And when making those judgement calls, the most important thing you can do is arm yourself with all the relevant variables. Know exactly what cards or combinations thereof that you’re looking for, and exactly what cards from your opponent’s deck are relevant to your decision, and the problem at hand becomes more straightforward. The ability to “see all the angles” may seem like an innate talent that some players have and weaker players don’t, but it’s a skill that you can develop over time if you’re willing to put the time in studying your games and those of others, talk through them with your friends and teammates, and learn from their perspectives.

If you have one player who is good at maximizing their own decks and one who is a student of the metagame and knows all the common sideboard cards from known decks, you should end up with two well-rounded players who can make these decisions with the same accuracy and precision as the pros.