### Disclaimer:

This article has a purely mathematical approach, but don’t be scared! There won’t be complex formulas nor endless calculations. Just some few plots and basic concepts that will help us to foresee what is waiting for us in the future of such an interesting format like Legacy.

### The problem:

The metagame is changing. Miracles, which once was considered one huge meta police, which maintained degenerated unfair decks at bay, is now gone. In such scenario, there is a real concern on the legacy community about how to deal with the combo problem. Such problem gets worse for some players that opt to play decks without blue, and consequently, without Force of Will, which is one of the best ways to battle against “unfair decks”.

On the other way, such dreadful scenario, where combos would run free and the chaos would reign, is not happening as expected. Loads of previous articles have being reporting that between the 5 main Magic archetypes (Control; Midrange; Agro; Tempo and Combo), the last one on the list have not been getting as many top 8s as expected, and almost no one have been seen on the top of the pod. Studies from Eternalmagic.com.br reported that combo decks have less than 29% of appearance on top 8’s on major events, both in Brazil (CLC, Alpha legacy, Liga mineira de legacy), and U.S. (Qualifies GP Vegas; 4k Channel Fireball, Classics SCG). As the opposite as one might think, the absence of Miracles on the metagame resulted in a greater appearance of Grixis Delver/Pyromancer archetypes, which have typically good matchups against most of the combo decks.

Despite this indicative based on the sample we have, it is common sense that when battling a combo deck, your success is highly dependent of the amount of hate cards you have on your sideboard. In other words, once “fair” decks have lots of cards on the main board focused on field interaction, most of them need to replace those cards for the second and third matches in order to better react to the combo threats. Some examples that can be highlighted as anti-combo hate cards are Flusterstorm; Surgical Extraction; Aethersworn Canonist and Sanctum Prelate. Being said that, we can conclude that ignore the presence of combos on the field is dangerous, and you can end up exposing yourself to extremely polarized matches.

As an aggravating point, we are very close one huge Legacy GP, which is probably the biggest events of the year. In such uncertain field, we have to as ourselves: How much should we actually fear and prepare our decks against combo, for such a big GP like Vegas? How fast those degenerated combos can be and how little interaction can they provide?

### Methodology:

In order to answer those questions, this article purposes one probability projection of an binomial model based on statistics extracted from Magic Online Championships on the week from 3rd to 10th of June. The Magic Online Sample was chosen due its great size, and also due the presence of some professional players on the field, which allowed us to make some statistically significant tests with one population similar of what is expected to be encountered inside a GP.

On simple terms, the aim of this quick study was to project, based on binomial distribution models, what is the probability of one player, inside one uncertain field (based on Magic Online), to be paired against one combo deck at least once during the 15 rounds of an GP tournament, and once paired, calculate the expected value for the match to end on the first few turns of the game. Basically, one binomial model is one way to estimate the chance of occurrence of certain predetermined result (in this case, be paired with a combo deck) within one certain number of random trials (in this case, during the 15 rounds of the tournament). See below the formula used for most of the calculations used on this article:

We can consider N the number of rounds on the GP (15); X the number of times we expect to be paired with an combo deck; p the probability of pairing with it on a single trial (generally the mean density of combo decks on the field), and q the inverse of this probability. For those that are not into math, you can simply ignore this part, and focus directly on the results.

To calculate the probability of pairing with a combo deck, the chance of occurrence of the event was measured through the sum of all combos identified on the sample divided by the total amount of decks. Once identified which decks can be considered combos, their speed in terms of how many turns they generally take to win the game – AKA: fundamental turn – was stipulated based on articles found on specialized forums related to those strategies. For instance, it was assumed that Show and Tell decks generally have their fundamental turn on 3, based on comments on MTG Salvation and on Channel Fireaball articles. In the same way, it was assumed the fundamental turn for the BR Reanimator as 1.

### Results:

#### General sample:

In the end, the analysis considered one total of 424 lists, being composed of 52 different deck strategies. By observing the distribution by archetypes, the results found are similar to the ones previously reported in other articles on this website. The 3 main archetypes had an overall uniform distribution floating around 30%, which is one indicative of a healthy format.

When we consider this result and project the probability of pairings within the 15 rounds of the GP, we assume that the probability for someone to face certain archetype in one single round is equivalent to its density on the sample. For example, on each round, we have 23,08% of chance to face an agro deck, and then we can assume that in every 4 matches, we might be paired against at least one of them. The graph below shows us the results of the projection of the probability of someone to be paired against each one of the archetypes at least once, during the GP. It’s worth noting that this analysis does not consider differences between archetypes in terms of win rate, so we are assuming that the proportion of those densities would maintain itself across all the rounds of the tournament, regardless how well their current standings.

By observing the plot above, we can assume that after all the 15 rounds, we have almost 100% of chance to face at least once each one of the main archetypes of the Legacy format. Taking into account only the combo archetype, the results suggest that the mean density of its representation is around 30,77% of the metagame, with one pairing probability of 99,6%, and one expectation of every player having around 5 combo matchups by the end of the event. In summary, the bottom line is: we cannot ignore the presence of combo decks in such a huge event like this. The numbers are not in favor of those who are expecting to be blessede by luck, hoping to dodge this kind of opponent. More fragile decks, with less tools to deal with combos, such as Goblins, Enchantress or even Burn might not be the ideal choice for this field.

The problem with these results is: different combos have different speed, and some decks might have distinct matchups based on it. On the next part of this article, we will focus on answering this specific question: once facing a combo deck, what to expect in terms of its speed to win the game, and how little will be our chances to interact with it?

#### Combos and expected value for fundamental turns:

Between the 52 different decks we found on this analysis, 16 of them were considered combo/unfair decks. As already said before, its density is around 30,77% and the data related to the fundamental turn of each deck, and their respective sample sizes can be seen on the table below.

By analyzing the descriptive statistics of this sample, we can infer that it behaves like one normal distribution (see histogram below), with mean = 2,613403 and variance = 0,780969.In other words, when we face a combo player (without knowing which specific deck he/she is playing), we can expect that he/she will finish the gamy between the second and fourth turn, but most likely on the third.

It is clear that most part of the combo decks focus on finishing the game on turn 3, some examples that can be highlighted are Show and Tell, Elves and Infect. Despites that, we also have one significant number (19) of combos specialized on going off on turn 1, such as BR Reanimator, Mono Red Sneak Atack and Belcher – without giving the opponents the opportunity to defend themselves (unless they have Force of Will). In the end, we can conclude that slower combo decks, such as Enchantress, UG Cloudpost and High Tide, which generally try to go off after turn 5 have simply disappeared from the meta. It is believed that those decks were better in long and grindy matches (such as, miracles), but fell off in favor when the fair decks stated to use huge amounts of had disruption and mana denial.

Considering expected value for fundamental combo turns, we can assume that when pairing blindly against someone in the tournament, our probability to receive a combo on the first two turns (crucial for hatebear-based decks such as Death and Taxes and Maverick) is relatively low (7,49%). The fundamental turns just start to be statistically relevant for the turns 3 and 4, which together composes over two thirds of the probability distribution. In theory, this results are in favor hatebear-based decks, which rely on cards like Thalia, Guardian of Thraben or Gaddock Teeg and Meddling Mage to stop unfair interactions. Once those cards generally hit the battlefield on turn 2, the overall perspective is that the most usual combo decks are not fast enough to race them. Even though, there will be always a chance of a bad pairing along the way.

With all these results in hand, we now must ask ourselves how this single probability is reflected through the 15 rounds of a tournament, always with the expectation to be prepared to every possible opponent we might face. Will such decks still be well positioned for an event like a GP? The graph below shows us the projection of the probability of one single person to receive a combo in different turns during the whole tournament.

When we consider the same probability distribution projected on the 15 rounds of the GP, it is evident that our assumption about the probability to face a very quick combo is the opposite. By the end of the event, the chance of someone to be paired against a turn 1 kill combo deck is around 54,28%. When considering the turns 3 and 4, this probability jumps over to 99,6%, which is threatening. This findings have direct impact in the way players might build their decks/sideboards in order to reach the top places of the tournament. If in order to reach top 8, one cannot lose more that 2 matches, it is evident that this person must be prepared for a 54% probability. With that in mind, this article recommends as its conclusion that, regardless the color of the deck of choice, one always have some sort of interaction in their main/sideboard that can be cast without need for lands in play (in other words, on turn 0). Cards such as Leyline of the Void/Santicity; Mindbreak Trap; Surgical Extraction and/or Faerie Macabre are essential sideboard pieces. Another point that must be highlighted is that there is no perspective for non interactive decks that do not aim to win the game after turn 3. Such deck choice would lead to an 99% of chance for an guaranteed loss to a combo deck during the event. Such scenario excludes value-oriented decks such as Post, Enchantress, or Goblins as good approaches for the metagame.

### Conclusion:

This article aimed to analyze the threat of combo decks on this new Legacy metagame, and ponder how much should we be prepared for this archetype on a big tournament. In order to achieve such goal, binomial probability models were used to calculat the projection and expected values for someone to face combo decks during the 15 rounds of a GP, and once pairing with them, what was the probability to be comboed off on the first turns of the game.

The results showed that despite the fact combo decks do not represent an overwhelming proportion of the Legacy metagame right now, its presence is big enough to not be ignored. Such findings affect directly on the deck and sideboard choices. In the end of this article, the authors recommend that every single deck must have plenty of wais in their main boards to deal with turn 3-4 combos, and at least on sideboard one way to interact on turn 0. By analyzing the archetype alone, it is not recommended to use any deck that is not able to present meaningful interaction or end the game on the first few turns of the match, due the high chance of someone to face this archetype on the rounds of one GP.

In the end, we must ponder about the limitations of this study. Once it considers generic abstractions for fundamental combo turns of different decks, which might not be one exact representation of the real probability for them to happen. The sample selected for the study was based on Magic Online, and might not be an exact representation of one Grand Prix metagame. This study does not considers the probability of the density of certain archetype to fall off, as the brackets on the tournament advances, assuming that they might maintain the same for all the 15 rounds.At last, this study uses parametric statistics for discrete samples, which might not be ideal (even though applicable), which may cause some discrepancies with same studies with larger sample sizes.