Jason Grabher-Meyer
12/14/2009

this article! - or -
Whenever you ask an experienced Duelist what they believe makes a successful competitor, you're bound to get a range of common answers. Practice, knowledge of a given metagame, experience with popular decks, good siding skills, the ability to read opponents, understanding of core theories like card presence and simplification, rulings knowledge… There are alot of factors that are widely recognized as contributing to a successful win record.

But one of the answers you don't hear often is “math”. In a game where those basic “+1s” and “-1s” so often hog the attention, real mathematical prowess can be underrated. After all, basic card presence is easy, right? And what do you need beyond that?

Well, by now you'll know there's a lot more to becoming a well-rounded Duelist than card presence theory alone, and when it comes to discussing the skills that competitive players value, the elephant in the room is hardcore probabilities. In a game where each draw can change the flow of the Duel, and a single topdeck can turn a loss into a win, it's shocking just how brushed aside basic probabilities are. Just this past week I've seen things that amount to mathematical heresy, coming from the mouths of experienced Duelists. A Twilight deck packing unnecessary copies of Wulf amongst a whopping eighteen potentially dead cards comes to mind. Or my favorite, an X-Saber player who ran four Lightsworn to get X-Sabers into his Graveyard, who then argued why one Charge of the Light Brigade was better than playing three.

That kind of stuff only happens when instinct and questionable logic replace black-and-white numbers.

How Does That Even Happen?
Probabilities and percentages are facts, and as such, they're incredibly convenient. If you want to know whether you should run two copies of a card instead of three, you can look at the Appropriate numbers and find your answer. There are a lot of elements in this game that can't be quantified and broken down to numerics. So why would anyone ignore the rare instances where the numbers are there, and everything could be so easy?

I think the issue is two-fold. First, the actual calculations can be time-consuming and not everybody knows how to do them. Second, a lot of people just hate math. Crunching those numbers is a lot like work, and it does take time, making it an unpopular choice for the average person who signed up to play a game (instead of, say, signing up to do their tax returns).

That's why I come to you today with an offering: a set of mathematical shortcuts that kicks all the work out of the window, filters out the more niche information, and breaks everything down to what I think are the ten most useful probabilities you could be using, but probably aren't. These numbers will help you build better decks, side deck more effectively, and will hone your off-table skills. They'll also help you make reads on your opponent, assist you in calculating risks on-the-fly during gameplay, and will make you a better Duelist when table-time finally arrives.

No work, no calculations – just ten numbers to memorize and then apply. Sound good? Then let's get to it.

39% - The Chance Of Opening With At Least 1 Copy Of A Card You Run 3 Of
When it comes to deck building, this might be the most important number you can learn. The above number (and the rest of the numbers we'll discuss here) assumes a 40-card deck, and when it comes to deck building, this might be the most important number you can learn. If your deck's win percentage goes up drastically because you opened with one particular card, then you need to know that running three copies means you'll open with that card in two out of every five Duels. If you don't open with one of your three target cards, this number increases by about 5% with each successive draw you make. That means you'll have a 39% chance of drawing one of your targets on turn 1, about a 44% chance of drawing that card on turn 2, a 49% chance of by turn 3, and so on.

Alternatively, 39% is also your opponent's chance of having Honest or Kalut when you attack; Blackwing - Shura the Blue Flame to punish your opening turn summon, or Gladiator Beast War Chariot to potentially set. Notice how this one number can be used in so many different ways. That idea – that one draw percentage can be used to find information in many different situations – is key to this discussion.

This 39% probability has several implications. First, it's mathematical proof that if you want to see an unsearchable card early, you should definitely run three copies. The fear of “well, I don't want to run three because then I might open with two when I only want one” is pretty irrational, and we'll have math for that a bit later. The alternative is running just two copies of a card you could be running in threes, which brings us to our next number.

28% - The Chance Of Opening With At Least 1 Of 2 Exact Cards
Hoping to see a card early on, but you only want to run two copies? This is what you're up against. While playing three of a target card will see you opening with a copy of it in two games out of every five, running just two copies means you'll only open with that card in about as few as one in four games. Perhaps more importantly, while the chance to draw one of your three target cards improved by 5% with each draw, the odds here are only going to grant you a 4% increase. You're starting off handicapped, and your progress is slower as well.

This number also serves as the probability of drawing any Semi-Limited card you run two of. If you want to know how reliably you can open with Allure of Darkness, Bottomless Trap Hole, Chaos Sorcerer, Judgment Dragon, or a non-Limited card commonly run in two's (think Mystic Tomato or Dimensional Alchemist), this is a good basic number to keep in mind.

As a sidenote, this number is important for more than just deckbuilding. Since this probability works for any card pairing you can imagine, you can use it to evaluate risk and calculate the odds of your opponent having certain plays. For instance, if you're considering an early-game over-extension committing multiple monsters to the field, but fear Mirror Force or Torrential Tribute, good news – you can use this number. Provided neither card is already in your opponent's graveyard, just add up the number of cards he or she has seen so far beyond their initial six, Multiply that by the 4% increase per draw, add that number to the base 28% and you'll have a pretty good idea of whether or not your opponent has drawn one of those two cards thus far.

Now keep in mind – just because your opponent has a high chance of having drawn a given trap, that doesn't mean that he or she has actually set it yet. The math can't tell you whether or not your opponent has made a particular play, but it can tell you the odds of that play being possible. Very useful stuff.

5.4% - The Chance of Opening With At Least 2 Copies Of The Same Card You Run 3 Of
On the flip side of the coin, what are the odds of you opening with two copies of a card you run in threes? What's the chance that you're going to be stuck with two Thunder Dragon, the odds that you'll pull off double Solar Recharge, or double Black Whirlwind?

The answer is 5.4% - about one in every twenty games. It's certainly possible to draw two copies of a card you run three of, but it's certainly nothing to be relied on. In addition, the fear of opening with two copies of the same card shouldn't be a big issue on anybody's mind, because even if that would somehow instantly cost you the game, we're only talking about one game in twenty – about one Duel in an entire ten-round tournament. The odds of drawing two of the same card only increase by about 2% on successive draws, so unless you play a lot of early game draw acceleration it's not a big problem. With that said though, by turn five (assuming only natural draws in your Draw Phase) you are looking at a 15% chance of seeing those two cards, so the numbers do slowly add up.

This number can also be applied when attempting to read your opponent's hand. If you see one Book of Moon, Charge of the Light Brigade, Honest, or Kalut early on, you can calculate the odds of seeing another. This is extremely handy in the first few turns of a Duel, when you're trying to get a rough idea of whether or not you should attack into a LIGHT monster or a Blackwing after being blocked once before by Honest or Kalut.

14% - The Chance Of Opening With a 2-Card Combo of 2 Different Cards You Run In 3's
Now we're getting a little bit more advanced. This probability represents the chance to open with a two-card combo of two different cards you're maxed on: for instance, Charge of the Light Brigade and Lumina, Lightsworn Summoner, or Black Whirlwind and Blackwing – Shura. If your deck is more creative and keys off a central two-card combo, this is how frequently you'll open with that play. It's not particularly reliable – the combo will appear in your first six cards in about one of every seven games you play – not even once every two matches. So you'd better get drawing!

The odds of drawing into your combo will increase by about 4.5-5% on each successive draw – about 23% on your eighth draw, and 33% by your tenth draw. Each “draw 2” card you play (like Destiny Draw or Allure of Darkness) is going to add another 10% to your chance to put together the card pairing you want. This number can help you evaluate both the viability of your own combo decks, plus how many draw cards you should run in them. It will also help you evaluate your opponent's odds of putting together particular combos you don't want to see.

49% - The Chance of Opening With At Least 1 of 4 Target Cards
Now we're getting to what I consider to be the good stuff! Playing two Bottomless, Mirror Force, and Torrential Tribute? This is the percentage chance you have of opening with one of your four defensive trap cards. It's a fifty/fifty chance, and it increases by about 5% with each draw until you draw your first defensive trap.

In addition, when you're evaluating the risk of making a summon and an attack in the early game, this tells you the odds of doing that safely – it's a coin toss. There's a definite possibility that at least one monster you summon could be destroyed this turn. If that would be a major problem, a “blind” Mystical Space Typhoon may actually be the smart play. If you have a secondary plan with which to test the waters though, tossing it out there and seeing if your opponent has the fifty/fifty chance to blow you off the table may be smart. Summoning Jain, Lightsworn Paladin, before going for Lumina and Garoth can protect your bigger play from a present threat. The same can be said for summoning Blackwing - Bora the Spear before Shura, or pressing with Zombie Master before Mezuki (or vice-versa depending on what you're expecting).

Again, remember: just because your opponent has the Bottomless Trap Hole doesn't mean he or she will activate it on the first monster you throw out – this is especially true in the case of Torrential Tribute and Mirror Force (cards Duelists like to hang onto until they can eliminate multiple cards). We're talking about possibilities, not absolutes.

This number also represents the chance to open with a draw card if you play two Destiny Draw and two Allure of Darkness, or a piece of spell / trap removal if you run Mystical Space Typhoon, Heavy Storm, and two more cards like Lyla, Lightsworn Sorceress, Dust Tornado, or Malevolent Catastrophe. It can also be applied to side decking. If you side in four cards, you'll have a 49% chance of seeing one of them in your opening hand.

58% - The Chance Of Opening With At Least 1 Of 5 Target Cards
Want to take the big-hitter trap cards that tend to get held back out of the equation, and just calculate the odds of Book of Moon and Bottomless? This is the number you want. When it comes to disposable early-game monster manipulation, these two cards reign supreme, and any Duelist maxing out on both has a 58% chance of opening with one of the two. Successive draws will grow those odds by about 5-6% until the first Bottomless or Book is drawn.

Similar to the last number we looked at, this is also your chance to open with a draw card if you play three Destiny Draw, three Solar Recharge, or even three Trade-In, alongside Allure of Darkness. Also like the last probability, this represents the chance to open with at least one sided tech card if you rotate five cards from your side deck into your main. By devoting one third of your side deck to a particular matchup, you can have about a 70% chance of drawing one of those cards by turn 3.

Why do Lightsworn players see Judgment Dragon so often? It's because with two copies of the Dragon and three Beckoning Light, they start the game with what's basically a three-in-five chance of holding it. The odds just get better and better from that point forward, both for each draw they make, and each card they mill.

15% - The Chance of Opening With At Least 2 Of 5 Target Cards
Building off the previous probability which told us that we have a 58% chance of drawing one of five target cards in our opening hand, this number tells us that after we see that first Book, Bottomless, or sided tech card from the opponent there's only a 15% chance that one of their remaining cards is another similar card. These odds will rise by 5-6% on successive draws though, so keep that in mind when you're evaluating risk. If you're packing five sided tech cards, use these numbers to figure your chances of hammering your opponent twice with stuff you sided in.

65% - The Chance of Opening With At Least 1 of 6 Target Cards
Solar Recharge and Charge of the Light Brigade? Creature Swappable monsters in a Zombie deck? One of your six Gadgets? This number may not be obviously useful at first, but it can be a real boon when you're putting a deck together. Not only can it tell you your odds of seeing a specific card you want, it can also tell you the odds of opening with a dead card if you happen to run 6 of them. Note that all previous “1 of [X] Target Cards” probabilities can do this too, for decks with fewer dead cards than six. This number increases by about 5% per successive draw.

On the tactical side, this number and the “1 of 5 Targets” number can help you decide your opening plays against decks like Macro Cosmos (which will run 5-6 copies of Cosmos / Dimensional Fissure), Blackwings (which will have 5-6 monsters of 1700 ATK or more before factoring in Kalut), and so on. It's the root of important information like “Better mill those Lightsworn as fast as possible!” and “Blackwings will have a better than two-in-three chance of attacking you (assuming they play Allure).” Nothing groundbreaking, but if you wanted to know the numbers behind the general play patterns, there they are.

15% - The Chance Of Opening With 1 Exact Card
AKA, “The Chance Of My Opponent Screwing Me With Heavy Storm If I Set Two Cards.” Any card you play one copy of will appear in your opening hand once in a little more than one of every seven games you play. This number increases by 2-3% – a 20% chance if your opponent opens with a single “draw 2” card or draws naturally for two turns (25% by the time your opponent draws four cards).

Strict math states that you might as well go ahead and set those two cards if your only concern is the pro Storm, but naturally, the mathematical risk and the impact (positive and negative) of your decisions need to be weighed and balanced very carefully. Losing two cards to one early on can definitely be devastating, but if you've seen top players like Jerry Wang or Fili Luna fearlessly setting two cards on their opening turn in feature matches without the backing of spell negation (which they've both done in the past), this is why. The 2-for-1 Heavy Storm is most certainly a frightening boogeyman, but it's bark can be a little bit worse than its actual percentage chance to bite.

This number also becomes very useful when you anticipate a 2-card threat, but eliminate the possibility of one of those cards. For instance, if I'm fearing Torrential Tribute and Mirror Force, I can take the base 28% chance of my opponent seeing either of those cards in the opening hand (one card from a two-card pairing as discussed earlier), modify that 28% chance for the number of draws my opponent has made, and get an idea of the chance that I could run into either of those traps. If I decide the benefits outweigh the risks and summon a second monster, and Torrential Tribute doesn't appear, I can disregard Torrential and then use this number (15%) to find the approximate chance of my opponent having Mirror Force. This can be the difference between “Attack with Lumina and Lyla and lose them both”, versus “Activate Lyla's effect, destroy Mirror Force, attack with Lumina.”

1.9% - The Chance Of Opening With 2 Exact Cards
Our last number is more the basis for axioms than something you should memorize. This one represents the odds of opening with double Judgment Dragon, double Goblin Zombie, or a pairing of Limited cards like Sangan and Torrential Tribute. Suffice to say the odds of opening with a specific pair of cards, each of which appears only once in your deck, is bad. The odds of drawing into both target cards doesn't increase much as turns pass, either – about .8% per successive draw.

For deck building this means some obvious things: don't rely on combos of Limited cards (well, duh), and if you only play two copies of a given card, don't expect to see both copies in a single game unless it's excruciatingly long.

On the tactical side this number is far more interesting. First, it can help in making reads: if your opponent activates Bottomless Trap Hole, you know you probably won't see the second one for a while. If you remove your opponent's Mezuki from play, the odds of them naturally drawing into their second is extremely low. However, what I find a lot more interesting is the level of opportunity that these one-in-fifty draws create.

See, a knowledgeable player knows the math here, and understands the reads I just touched on. That's a large part of what makes Sangan / Torrential so dangerous – it's not just that the play grants a search and a quick 1-for-1 or +1. It's that there's no real way an experienced player can anticipate it, because the odds are stacked so harshly against Sangan and Torrential Tribute appearing in an opening hand.

That means two things. First, it means that when an opportunity to make this kind of play presents itself, you should take it because it only comes around once in every five or six tournaments. In addition, you should use the scarcity of a two-specific-cards hand to your advantage. If you draw double Judgment Dragon, consider playing or discarding the first Dragon a bit earlier than you would if you only had one copy – it can lull your opponent into making the wrong read and over-extending, so you can punish them with the second Dragon. Learn to recognize when you've drawn a hand that should be this mathematically difficult to read. If you can set a trap by making what appears to be a weak play, but is actually strong due to your lucky hand, go for it.

And That's That
Committing these ten numbers and their successive draw percentages to memory isn't something you can do flawlessly over night. But if you start using these numbers during deck building to guide some of your tougher calls, and memorize a few that you think might be the most useful during your Duels, you can start putting them to work. The more you use them, the easier they are to recall.

Remember that while I listed uses for each of these numbers, every one is a tool limited only by your ingenuity. Think about the situations you find yourself in, and figure out different ways you can apply the numbers I discussed. Memorizing everything is just the first step – figuring out all the different ways to use them is the real skill you'll be developing. Master the basics, and you'll be able to use these numbers on the fly in situations that didn't even occur to you in advance.

Whether you memorize them, master them, or do neither and only apply these numbers to your deck building, each can boost your game and deliver results. A lot of this game is about gut, instinct, and guesswork. But some of it isn't, and if you can tell the difference and make the math work for you, you'll become a far better Duelist.

-Jason Grabher-Meyer