In this article, I guess I'll be trying to explain how.
Now, Yu-Gi-Oh! is essentially a game about plusses. "But it's about the life points or Exodia!" you say. Not really, actually. If you look at how most duels end, it ends with the winner having a lot of card advantage and the loser having very little (close to 0). In really close games, it usually comes down to top-decking, meaning neither player has much card advantage.
Even if card advantage isn't usually an indicator of who's winning, gaining plusses over your opponent is always nice. It gives them less outs to your plays or gives you more outs to theirs.
If you look at all of the current top-tier decks, you'll see it going on, too. Wind-Ups are all about discarding cards from the hand (so gaining card advantage because you're making your opponent go negative). Inzektors are the same thing: when they swarm from the deck to the field, they plus, while many of their effects also destroy opponent's cards at the same time (giving opponents negatives). While Dino Rabbit isn't necessarily a plus at first, Evolzar Laggia (which is a net 0 to summon, thanks to Rescue Rabbit) becomes a -1 for your opponent simply by being on the field, as it's an insta-Solemn Warning on one of your opponent's cards or requires them to discard an Effect Veiler to work around it.
Even Gravekeeper's, which is anti-meta, is doing it as well. Necrovalley shuts down opponent's graveyard-targetting effects, creating -1s (or at the very least 0s). Not only that, the Gravekeeper's Descendant and Recruiter combo, which adds a Gravekeeper's to the hand [+1] after destroying an opponent's card [+1] by tributing a Gravekeeper's [-1], is a powerful way to consistently get +1.
As you can see, the key to winning Yu-Gi-Oh! is making sure that you get those plusses/increase your card advantage.
Card ratios and probability
The first thing a deck needs to get plusses is a good ratio of Monsters, Spells, and Traps. Without this, there's no way you will be able to get what you want, when you want it, so you can make those moves that will deplete your opponent's card advantage or add to yours.
For example, say you're running a deck with 15 Monsters and 25 Spells/Traps. Given a starting hand of 6 cards, this is what you'd calculate:
15/40 ± 2/3 * √[(15/40)(1 - 15/40)] = 0.375 ± 0.323 = 0.052 to 0.698
Basically, it means you will start with between 5.0% and 70.0% of your hand being Monster cards 90% of the time, which translates to around 0 to 4 cards. (If you'd like to understand the equation used to get this number, refer to statistics z-intervals.) This means that more often than not you'll actually be getting around 1 to 3 Monster cards in your starting hand (with more on the lower end of that interval). If your deck doesn't rely heavily on Monsters, that's great; you can go ahead and only use 15. But if your deck needs those Monsters to get going, 15 will not be enough.
The formula for finding out the amount of each card in your hand that you will start with 90% of the time is as follows:
probability of that type of card ± 2/3 x √[(probability of that type)(1 - probability of that type)]
Also, there's the probability of top-decking the cards you need. As covered in many articles, such as Master's right here, it basically means that if you want to have nice, consistent draws, you want to include as many copies of the card as possible. Don't include too many, though, or else you'll get dead draws.
This is one reason to put searcher cards in your deck such as Reinforcements of the Army or Emergency Teleport. Not only does the searcher card add to your probability of getting a certain card (when you draw the searcher card and then pull the card you need from the deck), it also decreases the number of cards in your deck so you'll have a higher probability of top-decking some other card you'll need in the future.
Combos and probability
"But what about combos?" you might ask. "I can do something really cool by using 3 cards!" Yes, synergy is always good in a Deck. But what good is it if you don't get any meaningful plusses off those combos, if you deplete your hand/field to summon a powerful monster only to have it get Smashing Ground'd right afterwards? Therefore, when constructing combos, you want to keep in mind the probability that you'll be able to plus later.
(For the purposes of this article, I won't be looking at comboing with the Graveyard. Most Graveyard effects are usually a net 0: for example, Plaguespreader Zombie summons itself [+1] but needs a card in the hand placed back onto the Deck [-1]. For Call of the Haunted, you use Call of the Haunted [-1] to Summon a Monster back onto the field [+1].)
When you're thinking about including combos in your deck that will give you plusses, you should be keeping in mind the probability of plussing with that combo. That's what makes a Deck efficient: the best ones can do it using the starting hand 25% of the time, at the very least. For example, let's say you need 3 cards to do this really cool combo that can generate plusses later. Assuming a perfect situation (where the slate is completely clean at the beginning of the duel), the chance of getting that 3-card combo is:
1 - (31C6 + 3 x 3C1 x 31C5 + 3 x 3C1 x 3C1 x 31C4)/40C6 = 18.8%.
Complicated, eh? If you want to see why those numbers are the way they are, click on the spoiler below.
If you don’t want to do deal with that, this is what the equation is essentially saying:
1 - (probability of getting none of the 3 cards + probability of getting only one copy of one card + probability of getting one copy each of 2 cards) = probability of getting at least 1 of each card
It gets worse if you only have 2 of each card in that combo, becoming:
1 - (34C6 + 3 x 2C1 x 34C5 + 3 x 2C1 x 2C1 x 34C4)/40C6 = 7.0%.And one of each card?
1 - (37C6 + 3 x 1C1 x 37C5 + 3 x 1C1 x 1C1 x 37C4)/40C6 = 0.2%.
Basically, if you’re trying to consistently pull off a 3-card combo with less than 3 copies of each of those cards in the deck, don’t. And even if you are planning on running the combo and have 3 copies of each card in your deck, it better have a darn good chance at gaining major plusses, because it’s still a pretty low chance that it’ll occur during a duel.
For that reason, the metagame is moving more and more towards single cards that can "combo" on their own, such as Tour Guide of the Underworld or Rescue Rabbit. Because you only need 1 card to start the "combo," the chance of effectively starting with it rises to (assuming you have 3 Tour Guides/Rescue Rabbits in the deck):
1 - 37C6/40C6 = 39.4%.
Add a few searchers and that percentage rises dramatically.
And those Dino Rabbit decks? They want to start with either a Tour Guide or a Rescue Rabbit in their hand. What’re the odds of that happening?
1 - 34C6/40C6 = 65.0%.
Two words: Holy. Crap.
That’s pretty darn consistent.
Needing fewer and fewer cards has been the case with Special Summoning trends as well, from Fusion and Ritual Summoning (1 specific card + 2 semi-specific Monsters) to Synchro Summoning (1 specific type of Monster and 1 other Monster) to Xyz Summoning (2 semi-specific Monsters). With each new iteration, it's becoming easier to combo into those Monsters.
Situational cards and probability
Ever had a card that you have your deck because it saved your butt once? I'm sorry to tell you this, but you probably should get rid of it.
Think about it this way. For this example, we'll use Judgment of Anubis. Anubis is a good card, no doubt about it, which is why some Duelists try to run it. However, think about how many times it really can be used. For the most part, your opponent will main a Heavy Storm and 2 MSTs, which are the only common things that will activate Anubis.
Assume you'll main 2 Judgment of Anubises, let's take a look at the probabilities involved. First off, your opponent has a 3 in 40 chance of getting a card that will activate Judgment of Anubis in their starting hand, which is:
1 - 37C6/40C6 = 39.4%.And since you have 2 of these cards, there's a 2 in 40 chance you'll start with it and actually be able to Set it facedown onto the field for use, which is:
1 - 38C6/40C6 = 28.1%.These numbers on its own makes the card very hard to play, because the probability that you'll have it Set on the field for activation by your opponent is 0.394 x 0.281 = 11.1%.
Of course, that probability will increase as the game goes on, because you’ll have that Anubis sitting there facedown unless it gets destroyed by a Trap Card/Monster effect. But only slightly. Even you draw an Anubis on your first turn and your opponent plays a card that would activate it 3 turns after that, the chance of that encounter would still only be 0.545 x 0.162 = 15.3%.
Do you really want a card that will only be used to block a Heavy Storm/MST that infrequently? Not only that, Judgment of Anubis is a virtual net 0 when you activate the card: you'll lose 2 cards (the Trap Card itself and the discard) and your opponent will lose two cards (the MST/Heavy Storm and hopefully a monster). In all other occasions, Anubis will be sitting dead in your hand or on the field, making it a virtual -1 for you.
(As stated above, Life Points matter very little compared to card advantage in a Duel, which is why I'm not counting it as a plus. There may be cases of not being activate a Solemn Warning because of a lack of life points, but those situations occur rarely.)
A card that's only going to be a net 0 most of the time and a -1 in all others? You should get rid of Anubis in this case. Replace it with a card that will make your deck more consistent.
That's not to say situational cards shouldn't be included at all. There are some great examples of those that you should run in your Side Deck, if not your Main Deck, like Chain Disappearance, D.D. Crow, Swallow Flip, and Maxx "C". But there's a reason why they should be in the Side Deck: against certain decks that are pretty common, the probability of using those cards against your opponent becomes so high it's almost a no-brainer to run them.
For example, take Starlight Road. There are now officially 5 very common targets that will activate a Starlight Road: 2 Torrential Tributes, 1 Mirror Force, 1 Heavy Storm, and 1 Dark Hole. And don’t forget about other cards that can easily activate the card, such as Black Rose Dragon, Gladiator Beast Gyzarus, or Judgment Dragon, just to mention a few. That means your opponent might have something that can activate your Starlight Road 5 times out of the 40 cards in their deck, making the probability of at least one of those being in their starting hand:
1 - 35C6/40C6 = 57.7%.
With 2 Starlight Roads in the deck, you'll be able to get it into your starting hand (and onto the field) 28.1% of the time. Therefore, that card will stop an opponent's card 0.577 x 0.281 = 16.1% of the time, assuming you Set it your first turn and they play a card that would activate Starlight Road on their first turn. That's better odds.
And, like always, the odds will only go up: Setting it on your first turn and then waiting 3 turns before your opponent plays something to activate it has a 0.742 x 0.281 = 20.8% chance of occuring.
Not only that, with so many cards that can activate it, Starlight Road has a smaller probability of being a dead card on the field (a -1). Add in the fact that Starlight Road is practically a +1 over your opponent (for your card [-1], you negate their card and destroy it [+1], plus you get a free Stardust Dragon [+1]), can potentially save you from big negatives or momentum-stopping moves, and give you piece of mind when attacking against a facedown (that could possibly be Mirror Force), Starlight Road seems like a pretty good candidate for a deck, or at least the Side Deck.
As you can see here, probability does play a big part in deck-building, especially when you've got all the right cards but still need to tweak the amounts of each or decide which card to tech over another. It's probability that helps those top tier decks plus often during games and end up winning.
Next time you're making a new deck, be sure to keep probability in mind. It might just make the difference between a barely-competitive deck and one that'll be tough to beat.
Edited by Mattimis, 11 March 2012 - 03:32 PM.