Using Numbers to Guide Deck Building

ARG Circuit JohnnyUsing Numbers to Guide Deck Building

Hi duelists!  Recently I visited locals, which I haven’t done in quite some time.  A few readers came up and said hi, which I always appreciate.  One posed this question to me: How is statistics relevant to Yugioh?  I couldn’t give an answer - not because I had nothing to say, but because there is just too much to say.  This kind of question is akin to “How do you play Yugioh?”  It’s very broad and has all kinds of different answers and applications.  Knowing how to collect and use data about the playerbase, about tournament results, about probability, and so on is all very useful.  While I can’t answer the entire question all at once, I can break it down into something more specific: How is statistics relevant to deck construction?

I’m going to show you why the numbers matter using 3-Axis Fire Fist as my example for this article.  Some decks are not very complicated and have ratios that are too straightforward to write about (Evilswarm).  Other decks are more flexible, where different ratios can do different but viable things to your strategy.  3-Axis is one of those decks.  There are so many different ratios people play, and people often disagree on which is best.

 Brotherhood of the Fire Fist - Spirit

Two Stages

The first thing I want to clarify about building this deck is that I see its construction taking place in two stages.  One is building your deck to do what it does best in a vacuum.  You want to have the best hands possible, the best draws possible, and the most consistent strategy you can get going for yourself.  The other stage is teching out card choices so that your deck can deal with the meta.  For example, if I put 2 [ccProd]Mirror Force[/ccProd] and 2 [ccProd]Dimensional Prison[/ccProd] in a Dino Rabbit deck, that may increase the chances of drawing a card to protect [ccProd]Evolzar Dolkka[/ccProd] (stage 1).  But should I do this if the current meta is dominated by Inzektors?  Probably not, because [ccProd]Inzektor Hornet[/ccProd] pops battle phase traps for free (stage 2).

The first stage is where my mind is the most on probability and ratios.  In the second stage, I try to adapt my deck as best as I can to the important matchups, without sacrificing from the ratios I chose in stage 1.  These stages blend together a lot of the time; they don’t necessarily take place in two distinct chronological events.  This article will focus on stage 1 of deckbuilding.

 Brotherhood of the Fire Fist - Horse Prince

Pairs, Pairs, and More Pairs

There are various ways to win with 3-Axis Fire, but there is one way that is more assured than anything else, and that is the Sacred Fire combo.  Sacred Fire is a combo that can be done going first or second, but is easier going first, that allows you to end your turn with a total of 8 or 9 cards.  If your opponent plays [ccProd]Maxx “C”[/ccProd], it is usually best to finish the combo anyway because you will still be in the lead in terms of card advantage and field presence after he draws 3 cards from your special summons.  The combo is simply to synchro summon with [ccProd]Brotherhood of the Fire Fist - Spirit[/ccProd] and [ccProd]Brotherhood of the Fire Fist - Leopard[/ccProd] or Spirit and [ccProd]Brotherhood of the Fire Fist - Rooster[/ccProd] into [ccProd]Brotherhood of the Fire Fist - Horse Prince[/ccProd], to use Horse Prince to summon Rooster (if you synchro with Leopard) or Leopard (if you synchro with Rooster, SOMETIMES), to use Rooster to search a Fire Fist monster for a follow-up play, and to use Rooster to swap a “useless” formation with a better one (such as [ccProd]Fire Formation - Tensen[/ccProd] or [ccProd]Fire Formation - Tenken[/ccProd]).  The combo can be initiated once Spirit and another level 3 is on your board.  There are many ways to have them on your board, due to card redundancies.  Redundancy is a very good thing in Yugioh.  For example, [ccProd]Fire Formation - Tenki[/ccProd] essentially gives you 4 copies of Spirit and 6 copies of Leopard.  Rooster’s formation swap effect lets you turn Tenki into the Tensu you need or Tensu into the Tenki you need.  Redundancies all around.

There are 15 possible pairs you can draw among these 5 key cards: Leopard, Tenki, Tensu, Rooster, and Spirit.  Out of these 15 pairs, 3 of them result in Sacred Fire-9 and 3 of them result in Sacred Fire-8.  The other 9 pairs do not result in the deadly combo, but can often still set you up for the combo next turn.  [ccProd]Rekindling[/ccProd] is also really helpful because it lets you continue the combo if Spirit’s effect is negated by [ccProd]Effect Veiler[/ccProd].  Lastly, you can add additional combinations by playing Foolish Burial, adding 3 more pairs to the mix, for a total of 9 possible pairs that lead to Sacred Fire.  For the record, there are no unique triplets that create the combo.  Any triplet that works is also a pair that works.

fire fist - rooster
Let’s call Leopard A, Tenki B, Tensu C, Rooster D, Spirit E, Foolish F, and Rekindling G.  If you open AB, AC, AE, BD, BE, CD, BF, EF, or FG, then you’re golden.  Cards like Leopard and Tenki are easy to decide on; they’re almost always a good draw, so you run the maximum amount.  Cards like Tensu, Rooster, and Foolish, which can be mediocre or worse going 2nd and in situations where you don’t have other combo cards, are more debatable.  To help decide on how many to run, we need to be able to answer the question: will I win more games from opening this than I will lose games from drawing this at inopportune times?  How can I tell how much a card benefits me without playing it a whole bunch?  Sometimes, all it takes is running the numbers.  What kind of numbers do I want to see?  Well, if playing additional copies of double-edged swords like Rooster and Tensu only help me marginally (say, 1%), then I may be better off just not running the max number.

Brotherhood of the Fire Fist - Leopard

Many of you have heard of hypergeometric calculator.  It lets you calculate the odds of drawing certain cards in your opening hand or when you have x cards left in your deck.

If I play 3 of a card, I have a 39.43% chance of opening it; 2 of a card, 28.07%; 1 of a card, 15.00%.  However, the numbers we want are a bit harder to find and cannot be calculated with hypergeometric calculator.  We want to know the odds of drawing pairs, not singles.  What if I want to find the probability of drawing Tenki and Leopard together?  I run 3 of each, and drawing them together means I get to combo, whereas drawing only one of them means I have to wait a turn and give my opponent the chance to set up defense.  Well, when do I NOT draw this pair?  I don’t draw this pair if I draw no Tenki, which is 37 choose 6 divided by 40 choose 6.  Nor do I draw this pair if I draw no Leopard, which is the same as the above.  100% = 1, so we have 1 - 2 *(37c6/40c6).  Sometimes, I draw neither Tenki nor Leopard, and I adjust for that by adding 34 choose 6 divided by 40 choose 6 to the formula.  The final formula, if you want to plug it into Excel, is 1-(2*COMBIN(37,6)/COMBIN(40,6))+COMBIN(34,6)/COMBIN(40,6).  The result is 13.9%.  This is pretty high for just one pair!  We have several more pairs to go.  However, to find the total amount, we can’t just add up all the results.  That would result in a lot of overlap, and even make it possible to go past 100% for some decks, which is impossible.  The actual calculation is really difficult, so my friend wrote a program in C++ to solve for all pairs.

Sacred Fire Program

The question: What are the odds I will draw at least one of these pairs? AB, AC, AE, BD, BE, CD, BF, EF, FG.  The answer:

If I run 2 Tensu and 2 Rooster, 36.54%.  If I run 2 Tensu and 3 Rooster OR 3 Tensu and 2 Rooster, 40.34%.  If I max out on Tensu and Rooster, 44.68%.  If I max out and also throw in [ccProd]Foolish Burial[/ccProd], 50.38%.  As you add more combo pieces, your chances of opening Sacred go up 3.8%, then 4.3%, then 5.7%.  Why does the biggest jump occur when I add Foolish Burial?  The answer is found in how many pairs it forms: a whopping 7!  Rekindling (3) + Tenki (3) + Spirit (1).

Foolish Burial
All these percentages are boosted a tiny bit more by running 3 Upstart.  It’s too complicated to calculate, though.  It’s not as simple as calling your deck 37 (so don’t go using 37 whenever you crunch numbers on stattrek).  This is because going through your deck, removing 3 [ccProd]Upstart Goblin[/ccProd]s, and drawing an opening hand is not the same thing as drawing an opening hand, and then playing Upstart if you draw it.



Deck Skeleton

Ok, so we know Foolish can be pretty dead in certain situations.  But so dead that you want to sacrifice a 5.7% increase in opening the mega-nuts?  In my opinion, no.  You can form your own conclusions about Rooster and Tensu.  There are valid arguments to both sides, but unless you have the numbers in front of you, you won’t be able to make a fully informed decision.  So I would build a 3-Axis deck starting with these 20 cards:

3 Brotherhood of the Fire Fist - Leopard
3 Brotherhood of the Fire Fist - Rooster
1 Brotherhood of the Fire Fist - Spirit
3 Upstart Goblin
3 Rekindling
3 Fire Formation -Tenki
3 Fire Formation -Tensu
1 Foolish Burial

But I also want to shove the fire down my opponent’s throat on turns other than turn 1.  Unless you’re playing 3.5-Axis, then you can achieve this goal by running more additional power plays with [ccProd]Onslaught of the Fire Kings[/ccProd] and [ccProd]Reckless Greed[/ccProd].  To make sure these plays go through, you’ll want cards like [ccProd]Fire Formation - Gyokkou[/ccProd] and possibly [ccProd]Mystical Space Typhoon[/ccProd].  Then it will be like you’re going first!

Onslaught of the Fire Kings
2 Onslaught of the Fire Kings
3 Reckless Greed
2 Gyokkou

Now we’re at 30 cards.  Rooster needs to search out Fire Fist monsters, so we’ll need [ccProd]Brotherhood of the Fire Fist - Bear[/ccProd] and [ccProd]Brotherhood of the Fire Fist - Gorilla[/ccProd].  This also means we’ll need some number of [ccProd]Coach Soldier Wolfbark[/ccProd].  Let’s just add the minimum for now.  Rooster also needs to search out at least one useful Fire Formation trap.  Tensen, Tenken, both?

2 Bear
1 Gorilla
1 Wolfbark
1 Tensen

Ta-da!  We’re at 35 cards that are a part of the “engine.”  Building from the bottom-up, we are now at a place where we can decide on things such as defensive cards or more synergy cards (such as [ccProd]Blaster, Dragon Ruler of Infernos[/ccProd] to go with Foolish).  This is the stage where one decides on tech for the meta as well.

Blaster, Dragon Ruler of Infernos


A thank you goes out to Richard for his work on C++.  He doesn’t play cards.  He’s a genius though.

Shout-outs to my friend Eric, for igniting my interest in fire and demonstrating a passion for learning and an intelligence for card theory that is truly motivating to a fellow duelist.

And a dedication to the readers who randomly say hi and remind me that I’m not talking to a brick wall when I write, laugh out loud.

The Circuit Series comes to wonderful Las Vegas, NV on March 15-16, 2014! Click the pic below for all the juicy deets!

Until next time,

Play Hard or Go Home.





Johnny Li

Johnny Li

Houston, TX
Johnny Li

Latest posts by Johnny Li (see all)



  • Steven Savelle

    Nice article Johnny glad I got to meet and talk to you at strike zone awhile back

  • Matthew Dimon

    Great Article! One comment though, you listed 35 cards, not 32….

    • Johnny Site Li

      Good catch, will fix

  • JJ Abrams

    I was expecting to see a guide on “Number” Xyzs after reading the title.

  • CalabiYau

    I once tried simulating a dragon ruler deck with upstart, seven star sword and cards of conconence in C++ to see if I could predict how long it was likely to take to draw a single card (I had sixth sense in mind). I found that if at the end of each turn, you make the deck size smaller by two cards (to account for dragon ruler searches), you got results that matched tests on YGOPro quite well. Trying to do that to find the liklihood of drawing a pair of cards must take a while

  • Cyrus

    Wow, I’m really impressed! This was a fantastic article. This has been something really interesting to me for a while. Great job Johnny!

  • saintzera

    can you tell me how many “Rekindling” do you use for the calculation ? I don’t really understand programming but it looks like to me that “Rekindling” has not been counted yet since the amount of “G” in the script are so many
    and somehow I still believed that running 3 Upstarts in 40 cards-deck is the same as running 37 cards-deck in terms of probability

    • Johnny Site Li

      3 Rekindling, and that is just a snapshot, not the entire program. 3 Upstarts doesn’t give you 37. Let’s say you draw Upstart as your 4th card. The first 3 cards you drew were 1 in 40, 1 in 39, and 1 in 38, respectively. Then, the 5th and 6th cards are 1 in 36 and 1 in 35. When you play that Upstart, you’re getting a draw of probability 1 in 34. Now let’s look at what you get if you could take out 3 Upstarts before you draw your opening hand (a true 37 card deck). Your first through sixth cards are in 1 in 37, 1 in 36, 1 in 35, and so on.

      • Gabriele

        i think you are wrong with the upstart maths. Here’s why: suppose you draw a card from a 40 card deck and want to calculate the probability of drawing a x card, that you run only one copy, while you use 3 upstarts, too. So P(x1), P(x2)… and P(u1)… are the probability of drawing the x card or upstart as 1st, 2nd draw. In this formula i assume that you play upstart as soon as you draw it:

        P_total = P(x1) + P(u1) * P(x2) + P(u1) * P(u2) * P(x3) + P(u1) * P(u2) * P(u3) * P(x4) = 1/40 + 3/40 * 1/39 + 3/40 * 2/39 * 1/38 + 3/40 * 2/39 * 1/38 * 1/37 = 1/37

        Sorry for my bad english and i hope i have been clear enough…

        • Gabriele

          anyway great article! works like this are really enlightining!

  • danielw

    So yes you framed this in a specific example of fire fist but these numbers were written about years ago by Jason grabher meyer

  • Paul Cooper

    Who the fuck cares

    • Nelson Morales

      People who want to get better at deck building?