# Variance: Counting More than Your Blessings

There are many reasons we attend Magic tournaments, whether they are large or small, far or near. Those reasons vary from the hope at making a payday to a simple love of the game. Regardless of your reasons, we all share a common bond in our love for the game, the friends we make along the way and the memories we create in both strange cities and familiar haunts. It’s an added bonus when those memories and adventures can make us better — if we take something from our journey, we’re better off than when we started. The lessons I learned from one of my adventures may help us all benefit in the long run.

**Scene:** Columbus, Ohio. Origins Gaming Fair. Star City Open Series, Standard Open.

Enter: Two of your friends, one close, another relatively new, playing under the cameras for a Top 8 berth. You observe from a quiet distance as Gerry Thompson baits Javier Arevalo into an unprofitable attack. Javier looks over his options and chooses to take the bait rather than advance his board state and ship the turn. Down comes Restoration Angel from Gerry’s grip to ambush the Drogskol Captain, sending the game quickly out of reach.

Fast-forward to between rounds. You see your buddy Javier chatting it up with Cedric Phillips and overhear the question, “What do you think I did wrong that turn?”

Cedric responds in his regularly fashion, “You should have just cast Lingering Souls. If he had the [Mana] Leak, then you attack into him knowing he doesn’t have the mana for the [Restoration] Angel. If he doesn’t leak it, you just ship the turn. Even if he plays angel at that point, he does it for minimal value and you advanced your board state.”

It’s an intelligent response to an intelligent question.

At this point, I chimed in by jokingly jabbing at my friend, “Yeah, but Gerry’s running so hot. Everything off the top of his deck is just gas, gas, gas …” It’s a continuation of a jovial conversation we had the night before. Cedric’s ears nearly bled. He looked furious and went on a tirade about the stupidity of that reasoning. “You have zero control over his draw steps,” “If that’s your mentality, you’ll never really win anything,” and other seemingly truths came from the animated man. And, given the context of what he knew from our conversation, all seemingly deserving truths at that.

Regardless of the joking intent of my initial statement, Cedric’s response got me thinking. How many people have the mentality they had zero chance because their opponent was ‘running hot’ or because they were ‘bricking’ and ‘running cold?’ How many wins and losses can we truly attribute to this phenomenon?

It’s a known reality Magic is a game with some level of variance. Players try to reduce the amount of variance in our decks by playing redundant copies of spells, running the minimum number of cards in our decks, playing cantrips and dig spells like Ponder and Brainstorm. By attempting to control variance on some level, we attempt to reduce the chance to simply lose the game because of randomness. But do we?

We must first define randomness, which requires looking at its polar opposite, absolute certainty. Mathematics has proven beyond reasonable doubt that absolute certainty is an absurd concept. Nothing is absolutely certain; everything comes and goes in degrees of certainty. Therefore, randomness defines everything. This is the defining factor of the general laws of randomness and the Fundamental Formula of Gambling.

By applying this mentality to Magic, we know our decks follow the rules of probability and general laws of randomness. We then (mis)think our draw steps/Brainstorms/Ponders and the like as random phenomena — as if there were nonrandom phenomena to begin with. The cards in your deck don’t have memory, but they can be analyzed mathematically with the purpose to win the game. In random sequence, the cards are numbers following probability and statistical rules. The rules can be broken down into two fundamental elements: degree of certainty (DC) and number of trials (N). Add the Cerberus of almighty randomness, standard deviation, and you have the following formula:

## N = (Log(1-DC))/(Log(1-p))

*N = the number of tries for an event of probability (p) to appear with a degree of certainty (DC).*

To apply this logic to our game we have to examine the knowns and unknowns. Let’s say I am Javier Arevalo in the aforementioned situation. I know the worst situation is to attack into a Restoration Angel and be blown out. By utilizing the formula above, I can determine what the degree of certainty is for this to occur.

Hypothetically, let’s say I examine the board state. Gerry has four cards in hand. I know three of these cards are not Restoration Angel because I saw Mana Leak, Glacial Fortress and Island from playing Gitaxian Probe last turn. That leaves one unknown card in hand. When I Probed him, he had 37 cards left in his deck. I know he hadn’t played a Restoration Angel but was running four copies in his deck. That means he has a 4 in 37 chance to draw that Angel on his draw step. However, I also know that he did not shuffle after casting Ponder last turn. That created a 4 in 33 chance (12.12 percent) that he drew the Restoration Angel in his next draw step. I can say Gerry is intelligent enough to know Restoration Angel is his best play if I decide to attack on my next attack step. Utilizing the formula above where N = 4 and p = 12.12 percent, I can say with just above 50 percent degree of certainty that Gerry has drawn Restoration Angel and I should not attack into it. The best play is to cast Lingering Souls and ship the turn.

I know not all of us have calculators for brains than can do complex mathematics. Hell, I had to break out my old TI-82 to figure out the DC above (though in the process I did find old ASCII programs that brought back fond memories) and there’s no way I could perform this level of mathematics on the spot. But that doesn’t mean it can’t be helpful at all. At the very minimum, the performance of cross mathematics to determine the percent chance of draw, can provide us a higher level of understanding and probability to win our matches. It provides a deeper breadth of information, given we acknowledge a lack of information to begin with … confusing, I know.

Here’s an example of my own on this last trip. I’m 5-1 and paired against a Wolf Run Ramp opponent. It’s Game 3 and he has three cards in hand. I know one is a Primeval Titan. On board he has Sphere of the Suns with two counters, Birds of Paradise, Primeval Titan, Kessig Wolf Run, Inkmoth Nexus and six more mana of various types. There are 29 cards left in his library. I have Sword of War and Peace equipped to a Restoration Angel and five mana. I am at 18 life and 3 poison while he is at seven life. In my hand are Amass the Components, Snapcaster Mage and Consecrated Sphinx. My graveyard consists of Dismember, Ponder, Mana Leak and two Gitaxian Probes left to be flashed back by Snapcaster Mage.

I cast Amass the Components expecting to untap and kill his Inkmoth Nexus by flashing back Dismember with Snapcaster Mage. Once attacks were declared, he casts Beast Within and takes three damage. Then he untaps, draws a land, places it directly into play, activates Inkmoth Nexus and swings for seven poison. Had I taken the time to do the math, I would have known casting Amass was likely the absolute worst play I could make.

Analyzing the situation in retrospect, this is what I know: I had seen no Beast Within in Game 1 but two copies in Game 2, so I know there are at least two copies in his 75. In Game 3, he had not played Beast Within. There are a total of 31 cards between his hand and library that are unknown. Because he chose not to play Primeval Titan the previous turn, dealing three poison instead, his primary game plan was clear. By simple cross-multiplication, I can determine he had a 9.67 percent chance there was a Beast Within in his hand and a 3.4 percent chance he would draw one, if they had been singletons. With two copies, it was 19.34 percent in hand and 6.8 percent chance to draw; with three copies, a 29.01 percent chance in hand and 10.2 percent to draw; with a full playset, a 38.68 percent chance in hand and 13.6 percent chance to draw.

At this point, I know at bare minimum he has a 26.8 percent chance to have his out. A betting man probably wouldn’t take those odds with the opportunity to increase his chances of success exponentially by simply slowing down. It turns out he had a full four copies in his deck. Had I stopped to analyze the situation, my day may have ended up very, very differently.

To come full circle, can someone ‘run hot’ or ‘brick’ or ‘run cold?’ They certainly can. If they understand how to control the levels of variance in their deck to a degree of certainty, whether consciously or not, chances are they’ll seem to ‘run hot.’ Cedric was absolutely right that we cannot control draw steps without in-game mechanics supporting it (ala effects such as Sealed Fate) and that ‘running hot’ was not an excuse for poor play on our part. But by understanding the causal relationship between the general laws of randomness, our draw steps and what the manipulation of our libraries truly does to the laws of probability, we can reduce variance. It also provides some level of Degree of Certainty as to what our opponents can and cannot do. Science has shown this is the closest to absolute certainty we can ever hope to achieve. Some like to call it soul reading… I just like to call it math.

Until next time!

Matt Eitel

@YaoMatt on Twitter

Tags: Math, Odds, Probability, Variance

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