WS players are typically told that running 16 level 0 characters is optimal. Level 0 characters can vary widely by set. Some decks run them for support, some for searching, some for bonding. In general, the level 0 game is to build up stock, try to get the opponent as close to exactly 5 damage, and create hand and board advantage.

We’ve all heard the same stories: “Oh man, every time I go below 16 level 0s, I just don’t draw any!” or “Dude, you’re not using 16? That’s crazy!” or even “What? 10? Are you out of your mind?”

I’m guilty of overcompensating – I at one point in my Mono-Green Madoka deck once had up to 18 level 0 characters!

So, if we do have 16 level 0 characters, what are the chances that we’ll see at least one in the opening turns? Let’s assume that we all want to attack with -something- on the first turn. Sometimes just to build stock, it’s a concession one has to make to throw out their support character to the front line and send it into the abyss for one stock. We can also assume that we have only 8 level 0 characters with stats that can hold their own for a turn, and determine our chances of drawing one of those in the opening turns. To do this, we’re going to need to use … MATH™.

What separates WS from other card games is its unique mulligan system. In other games, such as Magic: the Gathering, a mulligan is taken in the form of putting the entire hand back into the deck, and drawing one fewer card. In Cardfight!! Vanguard, cards are picked from the initial hand, reshuffled into the deck, and redrawn. In WS, players actually discard up to five of their initial five cards and draw until they have five cards in hand. With the draw phase (draw one) and clock phase (put a card from hand into clock, draw two), a player could potentially see a total of thirteen cards before their first main phase!

To determine the chances of drawing a certain card from a deck, we’ll have to use hypergeometric distribution. In drier terms, hypergeometric distribution is: (quote from Wikipedia) **a discrete probability distribution that describes the probability of k successes in n draws without replacement from a finite population of size N containing exactly K successes. **Stat Trek has a calculator that you can use to run quick percentages.

*For those that are spreadsheet-happy, the syntax: *

**HYPGEOM.DIST(sample_s,number_sample,population_s,number_pop,cumulative) **

*works on Microsoft Excel to perform the same function.*

Let’s take a look at the numbers on the Stat Trek calculator if we run a calculation for if we:

Have a 50 card deck

Draw a 5 card hand

Have 16 level 0 Characters in the deck

Are looking to draw at least 1 Level 0 character

Now let’s interpret the data. The probability of drawing exactly one level 0 character in the opening hand from a deck containing 16 of them, is 35%. But, that’s not really the most helpful number. The most helpful number is on the bottom, where it gives us the probability of drawing one or more level 0 characters in this 5 card hand – 86.867%. The chances of completely missing a level 0 in the opening draw is 13.1%, which is not a small number. But, what if we were to factor in the draw and clock steps, thus extending the sample size to 8 cards?

Suddenly, the chances of missing a level 0 drop to a very difficult 3%, and the chances of seeing one or more surges to 96.6%! We can use the percentages from the initial five cards, as well as the percentages after a full 5-card mulligan with draw + clock to find our range of probabilities.

We can see that having 16 level 0 characters is a pretty justified number. If we have somehow managed to draw a full hand and mulligan five cards, the chances of finding at least one level 0 is an enormous 98%. The chances of finding more than one is a very high 86%. It’s very difficult to get stuck without any plays early on. But what if we tweaked these numbers? How aggressive can we get with a level 0 lineup?

Let’s see what the probabilities are with a crazy number – 10.

If we go down to 10 level 0 characters, the chances of seeing one or more in the opening hand is a pretty shaky 68.9%. It’s over half, but it’s not stable by any means. If we should not see any in the first five and take a full mulligan, our chances look like this:

Looking at the bottom number, the chances of seeing at least one level 0 have spiked to a respectable 89%. However, it’s very likely that that single level 0 character is all that the player may see for that level. Notice the number right above it is the calculation for the chances of seeing greater than 1 level 0 – 57.88%. This means that in most games, this deck will see one level 0 character on its first turn, and in about half of those games, it will draw more than one.

For newer players, especially those who are only playing with English sets, there isn’t a very large incentive to run very few level 0s. No English set (Bakemonogatari will change this very soon) currently benefits from Experience, or having high level characters in the level zone. When choosing a level 0 lineup, 16 is not a number to stick to religiously. The difference between 15 and 16 for example, is very small, and the chances of losing a game due to that one missing level 0 character are so small, that if it should ever happen, one should probably buy a lottery ticket. And as we’ve seen, even an aggressive number such as 10 can see some kind of consistency by the numbers, on the back of the game’s mulligan system and generous card drawing mechanics.

Questions? Comments? Send an email to theninthcx AT gmail dot com!

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