Differences between combo decks requiring 2 cards and 3 cards

If you're an evil combo player, you may have wondered a few times why 2-card, 3-card, or 4-card combo have such different results. Everyone must agree that a 2-card combo is better than a 3-card combo and definitely better than a 4-card combo, but how much better?

An honorable mention first for 1-card combo just like Titan-shift decks. The probability of getting a Primeval Titan or Scapeshift or Summoner's Pact by turn 5 is 93%.

TWO-CARD COMBOS

Let's take some examples. In Standard we had the famous Copy cat combo composed of Guardian Felidar + Saheeli Rai. The combo consisted of blinking Saheeli with Felidar Guardian, so you can activate the planeswalker's ability once again to create a new Guardian, which blinks Saheeli and so on until you have endless cats on the field. This combo requires 2 cards and you only have 4 copies of each in your deck, how likely are you to combo until turn 4?

The combo consisted of playing Saheeli Rai on 3 and Felidar Guardian on 4. The probability that you draw the Saheeli Rai until turn 3 is best analyzed by the probability that you don't draw it : it depends you don't draw it in your starting hand or the next 3 draws. This means that in order not to draw Saheeli Rai, you have to draw one of the other 56 out of 60 cards you have in your deck on the first draw, then one of the 55 out of 59, then the other 54 out of 58 and then thereafter ... For a total of 10 cards drawn until turn 3, you cannot draw Saheeli Rai. The probability of you drawing is 1 minus the probability of not drawing.

What if we consider that we are drawing more cards?

The probability of having the combo in the starting hand is very small, but as you draw more, the probability can be almost 50%.

And in Modern?

Standard has a more limited pool without so many tutors or cards with similar effects. So what happens if we see a 2 card combo in Modern? In Modern the accounts get a little weirder. Let's take the combo Ad Nauseam as an example. The purpose of the combo is to cast an Ad Nauseam after casting Angel's Grace or Phyrexian Unlife; draw all cards from your deck; exile 3 Simian Guide Spirit to produce 3 red mana and thus cast a Lightning Storm for much damage by discarding all lands.

The combo is only two cards because you only need Ad Nauseam + Angel's Grace or Phyrexian Unlife. But you don't necessarily need Ad Nauseam since you can pick it up from your deck with Spoils of the Vault. In total the deck has 3 Spoils of the Vault and 4 Ad Nauseam, totaling 7 cards copies of "Ad Nauseam"; and has 4 Angel's Grace and 4 Phyrexian Unlife, totaling 8 copies that allow you to combo without dying. So you want to draw at least 1 of 7 copies of Ad Naseaum and 1 of 8 copies of Angel's Grace.

In the end, the probability that you already have the starting hand combo is 40%, but after 8 draws the probability exceeds 80%. Remembering that the deck consists of a lot of card draws like Serum Visions and Sleight of Hand; This makes it one of Modern's most consistent combos.

THREE CARD COMBOS

Now imagine that you have a 3 card combo. The deck Devoted Company or Devoted Vizier for example can combo with Devoted Druid + Vizir of Remedies + Walking Ballista. Devoted Druid + Vizir of Remedies generate infinite mana and the Ballista ends up dealing infinite damage. The deck usually has 4 druids, 3 Vizirs and only 1 Ballista; how likely are you to get at least 1 of each in the field?

At most you can get around 15% chance of combo if you draw a lot of cards. These are not very encouraging numbers, right? But the deck was made and adapted for the combo. To be sure, as there are many types of decks with Devoted Druid, let's talk about the list below.

In this version you do not need the Walking Ballista if you have Duskwatch Recruiter // Krallenhorde Howler in hand, as with infinite mana he will fetch the ballista. 1 Shalai, Voice of Plenty is another Ballista substitute for winning with infinite mana; while 4 Chord of Calling can fetch you any missing cards from the combo. 2 Ranger-Captain of Eos are there to tutor Ballista also if necessary. In total you have 8 copies of Devoted Druid, 8 copies of Vizier of Remedies and 9 copies of Walking Ballista in the deck.

As the deck was set up, the probability of comboing until turn 5 is similar to the Ad Nauseam deck. The downside is that this deck has more ways for your opponent to interact with it; The advantage is that he can win games in other ways if necessary.

What if we focus all our strength in the combo God?

This version is similar to the previous one with the addition of cards like Finale of Devastation which also allows you to win the game using infinite mana. Both 4 Eladamri's Call and 4 Finale of Devastation, 3 Eldritch Evolution and 3 Neoform can fetch our combo cards. This means that we have at our disposal 18 copies of Devoted Druid, 17 copies of Vizir of Remedies and 18 copies of Walking Ballista in the deck.

The deck is so, but so centered in comboing that the probability of having the combo is almost 100% after just a few draws.

CONSIDERATIONS

The calculations here counted the probability that you have 1 or more combo cards, with a bit of a defect in the calculation with tutors serving the same cards, although this does not practically affect the final calculation.

These calculations did not count the interaction of the opponent either, nor if you can play the combo: you may not draw lands, but have the combo in hand for example. Plus, it doesn't count more complex moves like popping fetch to increase the likelihood of drawing a combo card.

Also, to simplify the calculations we considered zero the correlations between two probabilities of at least one card each; if counted the correlation, the correct would be using Hypergeomatric functions. This can cause a margin final error of 1% in the output.

CONCLUSION

What I would like to conclude from this article is that the consistency of the combo depends much more on the number of tutors in your deck than the amount of cards it needs. In an environment like Standard it is very difficult to make 3-card combo decks consistent, the chance of comboing will be around 20 to 30%. But tutors are plenty in formats like Modern, Legacy or Vintage. Combos in these formats may require 3 cards and still be "oppressive".