counter argument to the Trigger Phasing article

Hello.

First of all, I really like this blog. I've only discovered it last week, but I've already managed to read every article aside from a few deep clans.

I really liked every article so far, except the one about Trigger Phasing (for reference: <forum doesn't allow me to post links :( > ).
This article just doesn't seem like it was written by the same person, as it presents a few points without mathematical reasoning, and in my opinion very misleading points.

I have read the disclaimer, and I understand that keeping a "one trigger in every three cards" can be useful as a rule of thumb. However, I believe the following sentences in particular are 100% false:
- "This is why the "Shade" cards in Granblue screw up triggers so badly. It's not because they mill a lot, it's only 2; it's because they mill an even number. Milling a mod3=0 (divisible by 3 evenly) number is preferable to an even number due to the phasing principle."
- "If you draw, then soulcharge, then mill 2, don't get mad that a trigger or two became unusable and you set your drive checks out of whack."

These sentences suggest that an effect that mills or soulcharges (except in multiples of 3) is actually bad or changes your chance to pull triggers, and that may actually lead some people to avoid this kind of effects.


As an example, lets assume our deck has 42 cards left and 14 of those are triggers (exactly 1/3 but you can do the math with different numbers and achieve same results).
For this example I'm making a comparison between twin-driving or first milling 2 cards and then twin driving.

All calculations were made using hypergeometric probability with 15 decimal digits.


Scenario 1:
Twin-drive
42 cards in deck
14 triggers

Chance of 0 triggers: 0.439024390243903
Chance of 1 trigger: 0.455284552845528
Chance of 2 triggers: 0.105691056910569

Expected number of triggers in scenario 1:
0*0.439024390243903 + 1*0.455284552845528 + 2*0.105691056910569 = 0.666666666666666




Scenario 2:
Mill 2 cards (same chances as a twin-drive)
42 cards in deck
14 triggers

Chance of 0 triggers: 0.439024390243903
Chance of 1 trigger: 0.455284552845528
Chance of 2 triggers: 0.105691056910569

Now we consider the chances for a twin drive after each of these results.


Scenario 2.1:
Twin Drive after milling 0 triggers
40 cards in deck
14 triggers

Chance of 0 triggers: 0.416666666666667
Chance of 1 trigger: 0.466666666666666
Chance of 2 triggers: 0.116666666666667

Expected number of triggers in scenario 2.1:
0*0.416666666666667 + 1*0.466666666666666 + 2*0.116666666666667 = 0.7


Scenario 2.2:
Twin Drive after milling 1 trigger
40 cards in deck
13 triggers

Chance of 0 triggers: 0.45
Chance of 1 trigger: 0.45
Chance of 2 triggers: 0.1

Expected number of triggers in scenario 2.2:
0*0.45 + 1*0.45 + 2*0.1 = 0.65


Scenario 2.3:
Twin Drive after milling 2 triggers
40 cards in deck
12 triggers

Chance of 0 triggers: 0.484615384615385
Chance of 1 trigger: 0.430769230769231
Chance of 2 triggers: 0.084615384615384

Expected number of triggers in scenario 2.3:
0*0.484615384615385 + 1*0.430769230769231 + 2*0.084615384615384 = 0.599999999999999



Now we multiply the probability of scenarios 2.1, 2.2 and 2.3 by the expected number of triggers in each to get the "global" expected number of triggers in scenario 2.

Expected number of triggers in scenario 2:
0.439024390243903*0.7 + 0.455284552845528*0.65 + 0.105691056910569*0.599999999999999 = 0.666666666666666


So the expected number of triggers on a twin drive check with or without milling 2 cards from the deck is exactly the same, and there's no mathematical reasoning for "trigger phasing". I agree it can be useful as a guiding principle for quick decisions on the spot, but it should not be a reason to avoid any kind of effects.

Quote :

These sentences suggest that an effect that mills or soulcharges (except in multiples of 3) is actually bad or changes your chance to pull triggers, and that may actually lead some people to avoid this kind of effects.

That's because it does on average. The problem is that for some reason, you've chosen to ignore the principle of averages. In a small sample size, that may not happen, but on average over the course of many games, they can be an extra trigger-destroying force in your deck. Just like soulcharging. So no, it is not "100% false". It's true in a certain context--the context of the article.

When I get the opportunity to look over your math, I will, but I'll tell you right now that there are a great many things wrong with this article and that is not one of them. The biggest problem with the article is that it is forced to condense a lot of complex stuff into simple things for A. Usability and B. Because bushiroad made a stupid rule. If you want to take a grievance with the article, I suggest picking something that actually deserves it.

LightLightning wrote:

As an example, lets assume our deck has 42 cards left and 14 of those are triggers

Stop right there. You start with 50, starting vanguard is 1 and hand is 5 cards and you draw 1. That means there is 43 cards left. 43/50 * 16 triggers = 13.76 or 14 Triggers left on average.

42/50 * 16 = 13.44 or 13 Triggers on average.

So not only is it impossible to twin drive at this point, the numbers at this point is wrong.
However, I will continue on with these 'assumed' values.

Scenario 1:

Chance of 0 triggers: 0.439024390243903
Chance of 1 trigger: 0.455284552845528
Chance of 2 triggers: 0.105691056910569

Expected number of triggers in scenario 1:
0*0.439024390243903 + 1*0.455284552845528 + 2*0.105691056910569 = 0.666666666666666

Okay, so you have the expected number of triggers, but not the EXPECTED CHANCE to get a trigger. They are two different values. The chance to get at least one trigger is (again assuming these values) is ~56%. It's a small thing but it will make an impact. You can try twin driving about 200 times and see which result is more accurate.

I noticed that too just by skimming but I don't have time to comment on that or pull it apart while testing BT8, sorry. But thanks LFF for at least helping address the issue.

*Has changed the numbers from 14 triggers out of 42 cards to 12 triggers out of 36 cards, for hopefully obvious reasons.*

LightLightning wrote:

- "This is why the "Shade" cards in Granblue screw up triggers so badly. It's not because they mill a lot, it's only 2; it's because they mill an even number. Milling a mod3=0 (divisible by 3 evenly) number is preferable to an even number due to the phasing principle."

LittleFighterFox wrote:

You have the expected number of triggers, but not the EXPECTED CHANCE to get a trigger.

Because apparently the number of triggers you can expect to do isn't what we're looking for, here's probability:

---Probability of pulling triggers from Twin Drive---

2 triggers: (12/36)*(11/35)=11/105
1 trigger: [(12*24)/(36*35])*2=16/35
0 triggers: (24/36)*(23/35)= 46/105

a ~10.5% chance to pull 2 triggers, a ~45.7% chance to pull 1 trigger, and a ~43.8% chance to pull no triggers.

---Probability of pulling triggers after Ruin Shade's skill---

Probability of # of triggers being milled by Ruin Shade:

2 triggers are milled: 11/105
1 trigger is milled: 16/35
0 triggers are milled: 46/105

Probability of triggers after Evil Shade:

10 triggers out of 34
2 triggers: (10*9/34*33)=(5*3)/(17*11)=15/187
1 trigger: [(10*24)/(34*33)]*2=(10*8)/(17*11)=80/187
0 triggers: (24*23)/(34*33)=(4*23)*(17*11)/=92/187

11 triggers out of 34
2 triggers: (11*10)/(34/33)=(5)/(17*3)=5/51
1 trigger: 2*[(11*23)/(34*33)]=23/17*3=23/51
0 triggers: (23*22)/(34*33)=23/17*3=23/51

12 triggers out of 34
2 triggers: (12*11)/(34*33)=6/17*3=6/51
1 trigger: 2[(12*22)/(34/33)=(2*2*2)/(17)=8/17
0 triggers: (22*21)/(34/33)=7/17

Multiply probability of triggers getting milled or not with later resulting chance of pulling triggers(colors), and add up :
2 triggers: 15/1785+80/1785+92/1785=11/105
1 trigger: 16/357+368/1785+368/1785=16/35
0 triggers: 92/1785+368/1785+46/255=46/106

a ~10.5% chance of 2 triggers, a ~45.7% chance to pull 1 trigger, and a ~43.8% chance to pull 0 triggers after using Evil Shade.

SUDDENLY BOLIVIA

I think I just mathematically proved mill 2 shouldn't affect your ability to pull triggers...

Eh? You did? How does that work out?

Like this:

---Probability of pulling triggers from Twin Drive---
a ~10.5% chance to pull 2 triggers, a ~45.7% chance to pull 1 trigger, and a ~43.8% chance to pull no triggers.

has the exact same numbers as

---Probability of pulling triggers after Ruin Shade's skill---
a ~10.5% chance of 2 triggers, a ~45.7% chance to pull 1 trigger, and a ~43.8% chance to pull 0 triggers after using Evil Shade.

Alice wrote:

Quote :

These sentences suggest that an effect that mills or soulcharges (except in multiples of 3) is actually bad or changes your chance to pull triggers, and that may actually lead some people to avoid this kind of effects.

That's because it does on average. The problem is that for some reason, you've chosen to ignore the principle of averages. In a small sample size, that may not happen, but on average over the course of many games, they can be an extra trigger-destroying force in your deck. Just like soulcharging. So no, it is not "100% false". It's true in a certain context--the context of the article.

I didn't choose to ignore the principle of averages, I wanted to point out that there was no difference whatsoever from a mathematical point of view, so the difference on average will be none.

LightLightning wrote:

As an example, lets assume our deck has 42 cards left and 14 of those are triggers

Stop right there. You start with 50, starting vanguard is 1 and hand is 5 cards and you draw 1. That means there is 43 cards left. 43/50 * 16 triggers = 13.76 or 14 Triggers left on average.
[/quote]

I used those numbers from the top of my head, and yes, I agree those don't make a realistic scenario but, as I said, the result is the same with any situation you calculate. You can try it with 30 cards and only 2 triggers left on deck or whatever you like.

LittleFighterFox wrote:

Okay, so you have the expected number of triggers, but not the EXPECTED CHANCE to get a trigger. They are two different values. The chance to get at least one trigger is (again assuming these values) is ~56%. It's a small thing but it will make an impact. You can try twin driving about 200 times and see which result is more accurate.

Again, I used the expected number but the results would be the same.


In any shuffled deck, with any amount of triggers left, the probability of a trigger being the 1st, 2nd, 3rd, 4th, 5th or Nth card is exactly the same, how many triggers you pulled in the last N draws/checks is irrelevant.
So you can mill, soulcharge or draw as many cards as you like, the odds of you pullling 0, 1 or 2 triggers will, on average, be exactly the same.

Aww, my stoplight of a post gets no response?

TehNACHO wrote:

Aww, my stoplight of a post gets no response?

Not sure if you wanted a response from me or Alice.
If it was from me, all I can say is "thank you"... :)

I'm going to let you kids work this out. Call me when the final presentation is ready :V

I think I already showed all of what one needs to know about Evil Shade~

In short though, mill 2 should not drastically affect one's chances to pull triggers. In fact, the probability of pulling triggers overall(or you know, on average) should be the same.