Help @ Math probabilities (DA/TA/multihits)

[Masamune]

Hello,

I am trying to "combine" 3 incompatible random events called DoubleAttack, TripleAtt and Multi hits Weapon.

So far i managed to calculate each of these 3 random events individually (as shown in 1st attached image), but i failed to roll them all in one (2nd attached image is a -possible- tree of combined events i tried). Even tough it may have looked quite simple for me at first glance, i never managed to get :
- total chance=100%
- in the case of a player equipped with a Ridill, having DA=TA=0 firstly, should get 33,3333% for 1, 2 and 3 attacks in one round. Then if i add -for example- DA=9% to that player, i should get for 1 and 3 attacks 33.33%-9%/3=30.33%, and 33.33%+9%/3=36.33% for 2 attacks right ?

Could someone help me get a finger on eventual "translation" errors i may have done ?

Edit: 2nd image is with player equipped with a Kraken Club, ie Xmax=8.

[CDF]

One way to approach this is to assume double attack and triple attack traits are mutually exclusive. That is to say, one does not take priority over the other, or the DA/TA check occurs simultaneously, so that the sum of DA and TA rates may never exceed 1, which may not be true.

In the case of Ridill, let

D - double attack trait procs
T - triple attack trait procs
S - neither double attack nor triple attack trait procs (would be normal attack round w/o Ridill)

s_0 - Ridill doesn't proc (SA)
d_0 - Ridill DA
t_0 - Ridill TA

so,

P(single) = P(s_0)(1-P(D)-P(T))
P(double) = P(d_0)(1-P(D)-P(T)) + P(double|D)P(D) = P(d_0)(1-P(D)-P(T)) + P(D)
P(triple) = P(t_0)(1-P(D)-P(T)) + P(triple|T)P(T) = P(t_0)(1-P(D)-P(T)) + P(T)

Note the "ratio" of 1:2:3 hits from Ridill is considered 3:5:2, not 3:3:3.

[Masamune]

Are you assuming the multihits weapon chances don't enter into the SUM(D+T)=1 + "mutually exclusive" ?
That's THE info i voluntarily omitted since it is confusing me so much about * or + or nothing at all into the formulas...

Anyways, i just tested your proposition :

a)1st test
D=T=0
Xmax=3 (Ridill used)

P(single)=(1-(3-1)/3)*(1-0-0) = 1/3
P(double) = (1/3)*(1-0-0) + 0 = 1/3
P(triple) = (1/3)*(1-0-0) + 0 = 1/3
Ptotal=100%

Unless i made an error, that makes indeed a 3:3:3 for ratios 1:2:3attacks (i don't understand how and why you said 3:5:2)

b)2nd test
D=9%
T=0
Xmax=3 (Ridill used)

P(single)=(1-(3-1)/3)*(1-0.09-0) = 30.333%
P(double) = (1/3)*(1-0.09-0) + 0.09 = 39.333%
P(triple) = (1/3)*(1-0.09-0) + 0 = 30.333%
Ptotal=100%

Those results sound totally logical to me compared to 1st test

c)3rd test
D=T=0
Xmax=8 (Kraken Club used)

P(single)=(1-(8-1)/8)*(1-0-0) = 12.50%
P(double) = (1/8)*(1-0-0) + 0 = 12.50%
P(triple) = (1/8)*(1-0-0) + 0 = 12.50%
P(4) = (1/8)*(1-0-0) = 12.50%
P(5) = (1/8)*(1-0-0) = 12.50%
P(6) = (1/8)*(1-0-0) = 12.50%
P(7) = (1/8)*(1-0-0) = 12.50%
P(8) = (1/8)*(1-0-0) = 12.50%
Ptotal=100%

Then i put D=T=50% (that's what were making all my attempts fail when coupled with a multihits weapon, since total were getting either above or below 100%)
Let's see :

P(single)=(1-(8-1)/8)*(1-0.5-0.5) = 0%
P(double) = (1/8)*(1-0.5-0.5) + 0.5 = 50%
P(triple) = (1/8)*(1-0.5-0.5) + 0.5 = 50%
P(4) = (1/8)*(1-0.5-0.5) = 0%
P(5) = (1/8)*(1-0.5-0.5) = 0%
P(6) = (1/8)*(1-0.5-0.5) = 0%
P(7) = (1/8)*(1-0.5-0.5) = 0%
P(8) = (1/8)*(1-0.5-0.5) = 0%
Ptotal=100%

There we can see Psingle = 0% and all hits from Kclub only are nulled... doesnot make sense... and that was assuming Sum(D+T)=1 like you said.
Those results basically mean that someone using a Kclub and having very high Double AND Triple Attack rates would ALWAYS have 2 or 3 hits ONLY occuring in each round... I'm not sure if it's realistic ?

[Masamune]

Miscellaneous question : Is there a cap in game for (DA+TA) ?

(Something like D+T < 70%-80% for example would avoid the non-sense at the end of my previous post)

Edit: nvm, i made a quick research on the wiki and highest D+T i could get is a DRG/war with all possible DA equips + Fighter's roll(pt having a war) who 'd get D+T=60%+0%.
So looks like SE devs conceived items in such ways they dont even need a cap.

[CDF]

Your initial calculations showed you assume DA/TA traits override multi-hit weapon procs. So initial multi-hit weapon hit probabilities would not enter into sum(DA,TA) <= 1.

It has long been established that Ridill doesn't have a uniform distribution of hits. Studio Gobli reports the ratio to be 30:50:20 and independent experimentation provides support for that.

Also, Studio Gobli gives a hit distribution of Kraken Club as 5:15:25:25:15:10:3:2. for 1:2:...:8 hits (I have not been able to find any empirical evidence that confirms this distribution though.) Therefore, increased DA or TA will "nerf" KC (the expected number of hits based on the above distribution is 3.82, so any TA or DA will attenuate it). This "nerfing" would seem intuitive to many. So, theoretically, under our model of how DA/TA traits interact w/ multi-hit weapons, 50% DA and 50% TA will indeed cause Kclub to have always either 2 or 3 hits. Sure, 50% DA and 50% TA is totally unrealistic, but the result is consistent with the model.

As for a hypothesized cap on the sum of DA and TA, as you saw for yourself it seems impossible to check that at this time. One should check to see if summing DA and TA is valid, instead of DA coming before TA or vice versa.

(Aside: if DA and TA don't stack, that implies they are not independent events, and the easiest way to treat the relationship of DA and TA is to assume they are mutually exclusive as opposed to one overriding the other. Therefore, simply add DA and TA when considering adjustments to Ridill or Kclub and assume the DA/TA check occurs in "parallel." This relationship should be subject to verification, not that anyone really does since it is of limited interest, or I am just being willfully ignorant and not looking for the evidence.)

[Enedin]

Get your THF75 with Brutal out and go fight stuff. W/o merits, you might get 5% DA and 5% TA (10% TA with full merits), you might get something else. This is actually a nice test, I might do that.

I personally think they stack.

[Masamune]

er... excuse me CDF but , on contrary, i put all efforts to actually follow exactly SAME points of YOUR model, i assumed NOTHING, ie each of these 3 events can occur "in parallel".
The calculations which followed 1st post are actually following exactly your model and led to uniform ratio for multihit weap with D=T=0. So can you be more explicit where i "deviated" from your model when i went into numeric calcs please ?

For Enedin, i never seen in 4 years of playing a round with one non-jailer weapon swinging with BOTH DA and TA triggering. So what do you mean by "stacking"? for me obviously they do NOT (mutually exclusive)

if DA and TA don't stack, that implies they are not independent events, and the easiest way to treat the relationship of DA and TA is to assume they are mutually exclusive as opposed to one overriding the other.

I'm pretty sure they don't "stack" in the way as seen in game or @ wiki, which means all these 3 events are incompatible(or mutually exclusive like you said) between them. The definition of 2 incompatible events A,B is P(A intersection B)=0, which means also they can NOT be independant.
From there tough i tried to retrieve your model from theory, but ended up in something totally different... and false. Could you develop how you obtained P(single) P(double) and P(triple) please, so i can check where i made an error (Unless you find it yurself ^^).

[CDF]

It looks to me we basically agree on how DA trait interacts w/ multi-hit weapons (the traits proc before Ridill procs, and how the traits proc/don't proc affects Ridill).

The only major difference is that you were assuming a uniform distribution of hits for Ridill itself, before anything like DA or TA (w/ probabilities 1/3, 1/3, and 1/3 for 1, 2, and 3 hits respectively) when there is overwhelming evidence to the contrary.

We can obtain P(single), P(double), and P(triple) for Ridill by conditioning the final outcome, whether you observe 1, 2, or 3 hits using Ridill, on whether DA/TA procs occur beforehand (this can be easily extended to Kraken Club as you did). This is an example of the "law of total probability." I am just going to use my notation from last time as awkward as it is.

Example:

P(S) + P(D) + P(T) = 1, so
P(S) = 1 - P(D) - P(T)

P(single) = P(single|S)P(S) + P(single|D)P(D) + P(single|T)P(T)
          = P(single|S)(1 - P(D) - P(T)) + 0*P(D) + 0*P(T)
          = P(s_0)(1 - P(D) - P(T))

Here, we assume that if neither TA nor DA process, then Ridill will not proc (single attack) with probability P(s_0). Remember that "S" is the event that neither DA nor TA proc (normal attack round w/o Ridill).

P(double) = P(double|S)P(S) + P(double|D)P(D) + P(double|T)P(T)
          = P(double|S)(1 - P(D) - P(T)) + 1*P(D) + 0*P(T)
          = P(d_0)(1 - P(D) - P(T)) + P(D)

Here, if DA procs, then the outcome must be a double attack, or P(double|D) = 1. By similar reasoning, P(double|T) = 0 and P(double|S) = P(d_0), the DA proc rate for Ridill.

P(triple) = P(triple|S)P(S) + P(triple|D)P(D) + P(triple|T)P(T)
          = P(triple|S)(1 - P(D) - P(T)) + 0*P(D) + 1*P(T)
          = P(t_0)(1 - P(D) - P(T)) + P(T)

Here, if TA procs, then the outcome must be a triple attack, or P(triple|T) = 1. By similar reasoning, P(triple|D) = 0 and P(triple|S) = P(t_0), the TA proc rate for Ridill.

Of course, you can obtain these expressions "ad hoc" just by observing that DA/TA "converts" Ridill singles to DA/TA, DA "converts" Ridill triples to DA, and TA "converts" Ridill doubles to TA. This is all predicated on the unverified assumption that occurrences of DA and TA do not stack because these events are mutually exclusive. (If they stacked, they could be independent.)

[Masamune]

we assume that if neither TA nor DA process, then Ridill will not proc (single attack)

i wrote that :
"we assume that if neither TA nor DA process NOR Ridill proc, then single attack"
P(single)=P(Dbar)*P(Tbar)*P(Sbar)
P(double)=P(D)*P(Tbar)*P(Sbar)+P(S2)*P(Dbar)*P(Tba r)
etc...

with :
P(Dbar)=1-P(D) : chance of DA not procing
P(Tbar)=1-P(T) : chance of TA not procing
P(Sbar)=1-P(S2)-P(S3)-...-P(Sn), P(S1)=P(S2)=P(S3)=...=P(Sn)=1/Xmax in the case of a uniform distrib (thank you for yur explanation ^^) or P(S1)=1/3, P(S2)=1/5, P(S3)=1/2 in the case of Ridill.

But that reasoning doesnot give same results at all...

But i'm not sure about yur translation for event S...
1. For me i make distinction between 1 single hit and a multihit weapon which can deliver only "occasionnaly attack 2 to Xmax times"*
2. DA and TA are 2 totally different events, so why put them together in same event S ? Doing so would mean you actually consider the couple (DA;TA) has precedence over the event "multihit occurs", which i don't think and think more they are all 3 in same "level" ?

It looks to me we basically agree on how DA trait interacts w/ multi-hit weapons (the traits proc before Ridill procs, and how the traits proc/don't proc affects Ridill).

On that i'm not that sure actually, yes i'm agreed when trait don't proc, then multihit weapon get a proc chance and vice versa, but nowhere i saw any conclusive evidence that traits checks absolutely before multihit weapon, thats why i didnot assume at all since i don't know, and then put them all 3 (the 2 trait and multi hit) on same level.

What do you think?

[CDF]

Your probability statements seem to be based on the idea that Ridill processing, DA trait processing, and TA trait processing are mutually exclusive events in the same sample space. That would be probabilistically impossible because then the sum of the probabilities is then allowed to exceed 1. For example, if Ridill procs with a probabability 0.7, DA trait with probability 0.25 (possible), and TA trait with probability .1, then these events cannot be mutually exclusive (probabilities sum to 1.05, which does not make sense), unless SE imposes a cap on DA/TA rate in the presence of multi-hit weapons...

You wrote P(single)=P(Dbar)*P(Tbar)*P(Sbar), but this statement implies Dbar, Tbar, and Sbar are all mutually independent (and vice versa), but this is definitely not the case, as you were using the assumption that D, T, and S are mutually exclusive. If they are mutually exclusive, then they cannot be independent (and vice versa) and neither can the complements.

Yes, I considered DA and TA (from the traits, not Ridill) to be mutually exclusive events within the same sample space. I considered DA/TA to take precedence over Ridill's multi-hits based on that link to Talk:Ridill on FFXIclopedia, which has a bunch of data sets that are suggestive of that mechanism.

We literally cannot peer at the source code of FFXI and state affirmatively that this mechanism is true. But DA/TA "overriding" multi-hit weapon properties can be inferred from data, like "DA procs only on two hits of a WS" can be inferred from Penta Thrust data showing the distribution of hits. Well, in the latter case, the data was used to rule out the notion that DA procs on a WS are uncapped, so maybe that's more like a logical fallacy, but oh well.

[Masamune]

aaa ok i understand better now, indeed thats the problem i was encountering: Ptotal would be either lower or superior than 1.

Also thank you for info about DA/TA checking before multi hit check comes from data mining.

Ok i'll continue on that basis now, with jailer weapons this time...

Edit: btw, now that you gave me the real distrib for KC and Ridill, , im wondering if those for Joyeuse and Jailers weapons are indeed 1/2 1/2 ?

[Ultimagi]

I recently did some math to see if Brutal Earring was good to wear during Evisceration for a Thief. I'm pretty busy at work at the moment, but when I have some time I'll figure this one out.

[Ultimagi]

D = Double Attack
T = Triple Attack
O2 = Occasionally Attacks Twice
O3 = Occasionally Attacks 3 Times

Define the probability of each event occurring:

D = 0.1
T = 0
O2 = 0.5
O3 = 0.4

Define the probability of each event not occurring:

1 - D = 0.9
1 - T = 1
1 - O2 = 0.5
1 - O3 = 0.6

Now relate them in the order they are checked:

D = 0.1
T = (1 - D) * T
T = 0
O2 = (1 - T) * O2
O2 = 0.45
O3 = (1 - O2) * O3
O3 = 0.18

And don't forget to find the probability of nothing occurring:

N = (1- O2) * (1-O3)
N = 0.27

The sum of all possibilities will always equal 1:

0.1 + 0 + 0.45 + 0.18 + 0.27 = 1

If you want to figure out the expected number of attacks for an arbitrary attack round, multiply the probability of each occurrence by the number of resulting attacks, and sum them. With the numbers above, the expected number of attacks is 1.91.

If you define your variables and equations in a spreadsheet, you can alter the variables to see any changes that result. Say for instance you equip a Brutal Earring and fully merit Double Attack. D becomes 0.2, and the expected number of attacks increases to 1.92. The difference is negligible, for the Ridill. If you want to factor dual wielding, you now need to define:

N * N = 0.243
N * D = 0.09
N * T = 0
N * O2 = 0.405
N * O3 = 0.162
D * N = 0.027
D * D = 0.01
D * T = 0
D * O2 = 0.045
D * O3 = 0.018
T * N = 0
T * D = 0
T * O2 = 0
T * O3 = 0

Expected number of attacks is 3.01.
Brutal Earring and Double Attack merits increase to 3.12.

Edit:

You need to redefine N for the main hand weapon as (1 - D) * (1 - T) since it can't occasionally attack more than once.

[Masamune]

Interesting math, but i'm trying to avoid this as much i can :

Now relate them in the order they are checked:

D = 0.1
T = (1 - D) * T
T = 0
O2 = (1 - T) * O2
O2 = 0.45
O3 = (1 - O2) * O3
O3 = 0.18

...because we(at least me) don't know for sure if there s even a check between traits, only some approximative evidence about "traits checks MAY occure before multi hits weapons checks", that's only thing i conceded in CDF's model his way of putting DA/TA traits in a separate sample space than multihits's, but that's it; don't know anything about how DA check interacts with TA so put them both in same "level". So far results are ok, barring the wrong distribs i used but fixed now.

Atm i still didnot manage to combine Jailer weapons with CDF model ...
So far i 1st assumed, since Jweaps are weapons and not traits, could try to put them in same sample space as multihits, ie pretty much same model as CDF's except i added the following new cases :
- for 2 attacks: +Jweap
- for 3 attacks: +Jweap*DAonce
- for 4 attacks: +Jweap*(DA²)+Jweap*TAonce
- for 5 attacks: +Jweap*DAonce*TAonce
- for 6 attacks: +Jweap*(TA²)

..... again Ptotal different than 1 -.-

2ndly i tried with assuming Jweap can be considered as a trait, but then the "traits sample space" would be null as soon as D+T>50%, ie (1-J-D-T)<0 ...

[Masamune]

Btw i just remembered something about Ultimagi's post :

For WS, multihit weapons basically don't trigger at all, only DA/TA, and only on 1st hit + once on any additional hit after the 1st. When DWing, multihit weap or not doesnot matter, offhand weapon will add only ONE single hit in the end.
Example for Evisc: 5hits WS + 1 possible DAorTA on 1st hit + 1 possible DAorTA on one of the remaining 4 hits + 1 additional hit from offhand weap in the end => cap @ 8 hits with either 1TA or 2DAs, and all connected.

I don't remember where i found that and wasnot exactly worded like that, but yu get the idea.

[Ultimagi]

I'm pretty sure Double Attack is checked before Triple Attack. I know several Thief/Warrior who have tried it with max merits and such.

[Ultimagi]

My possibilities for Evisceration were;
NNNNN
DNNNN
DDNNN
DNDNN
DNNDN
DNNND
NDNNN
NDDNN
NDNDN
NDNND
NNDNN
NNDDN
NNDND
NNNDN
NNNDD
NNNND
TNNNN
NTNNN
NNTNN
NNNTN
NNNNT
And another set with a 6th attack for /Ninja. Double Attack and Triple Attack have been shown to occur on the offhand hit of a weaponskill, by the way.

[Masamune]

I'm pretty sure Double Attack is checked before Triple Attack. I know several Thief/Warrior who have tried it.

erm... can you explain me the modus operandi, because honestly i don't understand how one make the difference and can affirmatively say "DA checked true! omfg my TA got fucked!" please ?

or post a link to those tests, would save yu an explanation ^^

[Ultimagi]

With 15% Double Attack and 11% Triple Attack, Triple Attack parsed at approximately 9%.

[Ashokan]

With 15% Double Attack and 11% Triple Attack, Triple Attack parsed at approximately 9%.

How big was the sample?
What was DA parsed rate?