What is up everybody? SimplyJeff is back and today we are talking about everyone’s favorite subject in school: maths! Okay, while it wasn’t everyone’s favorite, it was one of my top classes for enjoyment. I’m shocked I never thought to do this before, but today I would like to put some math into Vanguard. Many people often complain that Vanguard is a luck based game, and while there may be some luck based elements, everything can be boiled down to simple numbers and percents. Since this article is mainly math, I’d like to apologize now if it is drier than my usual articles, ’tis the nature of the beast, and the beast is statistics. We got a lot to cover, so I’m just going to dive right in. I would like to thank Reddit user kumadown for suggesting this article. And, away we go!

*Establishing norms:*

Before we begin to really analyze the numbers themselves, we have to set some standards. A deck is extremely unique depending on the clan, so making a perfectly comprehensive list would be next to impossible. First I’d like to establish a norm for trigger spread. Below is the trigger spread for champion regionals decks and the top three decks at the Bushiroad World Championship 2015 in North America. (side note, I just find it really cool that a modern Machining deck placed first in Mexico)

**Regionals (* indicate frequency of occurrence) **

*7 crit 5 draw 0 stand 4 heal

****8 crit 4 draw 0 stand 4 heal

*5 crit 3 draw 4 stand 4 heal

**10 crit 2 draw 0 stand 4 heal

*9 crit 3 draw 0 stand 4 heal

*6 crit 2 draw 4 stand 4 heal

*4 crit 0 draw 4 stand 4 heal

**Championships**

7 crit 0 draw 5 stand 4 heal

10 crit 2 draw 0 stand 4 heal

8 crit 4 draw 0 stand 4 heal

*source: **http://cf-vanguard.com/en/cardlist/deckrecipe/*

I picked North America because that is where I live, but I honestly do not believe it will cause any major discrepancies. Now, it is important to note, the only builds running *10 crit 2 draw 0 stand 4 *heal spread are Shadow Paladins, but the *8 crit 4 draw 0 stand 4 heal* spread was used in multiple decks, so I will be using this as the norm. For the course of this article, a “normal deck” will be defined as containing 8 crit, 4 draw, 0 stand, and 4 heal. Following the same pattern for the remaining grades, I define a “normal deck” contains the following:

**Grade 0:**

**8 crit**

**4 draw**

**0 stand**

**4 heal**

**1 starter**

**Grade 1:**

**4 sentinel **

**10 other**

**Grade 2: 11**

**Grade 3: 8**

This ratio has been the same in Vanguard for awhile now, as it seems to provide the best spread. Now that the deck norm is established, I’d like to establish a few other rules before we get started here. First, I am going to assume no cards enter back into the deck or are superior called out of the deck to throw off the number of cards remaining. I am also going to assume that there were no extra draws. If I tried to factor in all that, it would make this article incredibly dense. Next, I am going to make a couple assumptions, based on games I’ve played with friends. Most games don’t see a major field presence until both players are at grade 3, which is when many cards can safely used their abilities. Therefore, I would like to state that the first attack of the game will only deal 1 damage with no crits. Once a player rides to grade 2, typically another column is called, so this attack will deal 2 damage with no crits. Lastly, once a player reaches grade 3 or 4 (depending on if they have the ability to stride right away) the full-on assault begins, a second column is called, and the attacks will deal 3 damage with no crits. Obviously shielding plays a factor in all this, but we’ll cross that bridge when we get there. So let’s recap!

-A normal deck contains the spread bolded above

-A “Grade 1 vanguard” turn attack will be able to deal 1 damage with no crits

-A “Grade 2 vanguard” turn attack will be able to deal 2 damage with no crits

-A “Grade 3+ vanguard” turn attack will be able to deal 3 damage with no crits

*What are we testing for:*

So now that all that is out of the way, what the heck are we trying to figure out? Well, I have a list of things I would like to analyze:

*-g-assist success*

*turn 1*

*turn 2*

-Crits

first turn, second turn, third turn no stride,

third turn stride

-Draw your preferred rides in opening hand

first try

While I would love calculate all of these, that would make this article far too long. So for today, I will only be looking at G-assist success rates.

Do note that “miracle heals” will not be calculated, as it is impossible to tell what the gamestate is when a miracle heal is needed. Simply take the number of heals you know are remaining in your deck, and divide by the number of cards remaining in your deck, and that should let you know the probability of a miracle heal. Also, grade 3 g-assists will not be calculated, as I’ve never seen anyone g-assist for a grade 3. I’m not saying it won’t ever be needed, but it’s so rare that I will not include it here. Well, I believe I established everything I need to in order to start crunching numbers, so let’s get to it!

** *DISCLAIMER* **So before we begin, I am only human and it is possible some of the math could be wrong. I tried my best with my understanding of statistics, but please don’t be afraid to let me know if something is not right. I do believe everything is correct though.

**FORMULA USED:**

The probability of drawing at least 1 specific grade in 5 draws is equal to 100% – the probability of not drawing any specific grade in 5 draws. We will define the probability of a successful g-assist as P(s) and an unsuccessful g-assist as P(u) *heheh P(u)….because they stink…* This is written as:

P(s) = 1-P(u)

P(u) = # of all possible combinations of 5 card draws where you will not see the specific grade divided by all possible combinations of 5 cards draws where you could see at least one of the specific grade. This is written as:

P(u) = C(n1,r)/C(n0,r)

n0 = number of all cards in deck at the time

n1 = number of all cards in deck that are ** NOT** the grade you are trying to find.

r = number of cards you will be drawing *since a g-assist always reveals 5 cards. r is always 5*

Therefore, we finally get to the equation we need to solve all these probabilities:

P(s) = 1- [C(n1,5)/C(n0,5)]

*G-assist success rate:*

**Turn 1:** Starting order does not affect turn 1, as no one has been able to deal damage, even if the second player has to G-assist.

Cards in deck: 43

Grade 1s remaining in deck: 14

Chance of getting a grade 1 in 5 draws= 1-[C(29,5)/C(43,5) = 87.66%

**Turn 2:** Listed below is every possible scenario you can find yourself in by turn 2, assuming no trigger effects, such as crits or stands.

**0 damage taken, 0 drive checks**

Cards in deck: 42

Grade 2s remaining in deck: 11

Chance of getting a grade 2 in 5 draws = 1-[C(31,5)/C(42,5)] = 80.03%

**B) 1 damage taken, 0 drive checks (damage is not grade 2)**

Cards in deck: 41

Grade 2s remaining in deck: 11

Chance of getting a grade 2 in 5 draws= 1-[C(30,5)/C(41,5)] = 83.25%

**C) 1 damage taken, 0 drive checks (damage is grade 2)**

Cards in deck: 41

Grade 2s remaining in deck: 10

Chance of getting a grade 2 in 5 draws= 1-[C(31,5)/C(41,5)] = 77.32%

**D) 1 damage taken, 1 drive check (damage is not grade 2)**

Cards in deck: 40

Grade 2s remaining in deck: 11

Chance of getting a grade 2 in 5 draws= 1-[C(29,5)/C(40,5)] = 81.95%

**E) 1 damage taken, 1 drive check (damage is grade 2)**

Cards in deck: 40

Grade 2s remaining in deck: 10

Chance of getting a grade 2 in 5 draws= 1-[C(30,5)/C(40,5)] = 78.34%

**F) 2 damage taken, 0 drive checks (no damage is grade 2)**

Cards in deck: 40

Grade 2s remaining in deck: 11

Chance of getting a grade 2 in 5 draws= 1-[C(29,5)/C(40,5)] = 81.95%

**G) 2 damage taken, 0 drive checks (1 damage is grade 2)**

Cards in deck: 40

Grade 2s remaining in deck: 10

Chance of getting a grade 2 in 5 draws= 1-[C(30,5)/C(40,5)] = 78.34%

**H) 2 damage taken, 0 drive checks (both damages are grade 2)**

Cards in deck: 40

Grade 2s remaining in deck: 9

Chance of getting a grade 2 in 5 draws= 1-[C(31,5)/C(40,5)] = 74.18%

**I) 2 damage taken, 1 drive check (no damage is grade 2)**

Cards in deck: 39

Grade 2s remaining in deck: 11

Chance of getting a grade 2 in 5 draws= 1-[C(28,5)/C(39,5)] = 82.93%

**J) 2 damage taken, 1 drive check (1 damage is grade 2)**

Cards in deck: 39

Grade 2s remaining in deck: 10

Chance of getting a grade 2 in 5 draws= 1-[C(29,5)/C(39,5)] = 79.37%

**K) 2 damage taken, 1 drive check (both damages are grade 2)**

Cards in deck: 39

Grade 2s remaining in deck: 9

Chance of getting a grade 2 in 5 draws= 1-[C(30,5)/C(39,5)] = 75.25%

As you can see, in almost every possible scenario, due to the amount of grade 1 and 2 cards in your deck, it is almost always a good idea to g-assist. Now, just because the probability may be high, you obviously aren’t guaranteed a successful g-assist. If you are worried about grades sticking together, always make sure to pile shuffle before any cardfight.

Well everyone, that is going to conclude this article. Sorry it was hefty, but I wanted to make sure it would be thorough. I have a few other ideas in the works, such as a comprehensive clan guide and a “how to pick your first clan.” Also, I have a video in the works where I will be discussing my favorite and least favorite metas of the game, such as Limit Break, Legion, and Stride. If there is anything other than those topics you wish to see, please let me know below!

*Well that’s it for this post. Once again feel free to contact me on reddit **here** or leave a reply on this thread for suggestions on what I should do next. I’ll try to get my posts out in a timely fashion, but do ask for understanding if I don’t.*

*SimplyJeff, signing off.*