# DoT Chance: a mathematical approach

I’ve been overhauling a lot of my threads so they are a little less arcane and more useful for general gameplay, this one is no different. In fact it was probably one of the worst offenders…

Anyway, this guide will feature a mathematical analysis of DoT/status effect chance, and how it is effected by chance increasing skills (more pep, flicker, fuel the fire and BAR), and by instances of damage (splash damage and secondary/multiple pellets). Additionally, there will be an analysis of what this actually means for gameplay, specifically slag weapons.

Have you ever asked yourself: “why is it that the Pimp is considered a super good slag weapon, while a Vladof assault rifle isn’t?” Well either way, you have now. So, there are 2 reasons:

1. the pimp just has a better DoT chance than the AR
2. the pimp has 7 instances of damage per shot, each of which has that increased DoT chance, leading to an incredibly high true chance.

But exactly how high?
That question can be answered by the following formula (for a derivation of this formula, refer to the bottom of this post):

P=1-(1-C)n

where:
P is the true DoT chance per shot
C is the listed (card) DoT chance
n is the number of damage instances (splash and/or pellet count)

DoT chance increasing skills modify C multiplicatively as in:

Cm=Cu*(1+M)

where:
Cm is the modified base DoT chance
Cu is the unmodified (card) DoT chance
M is the DoT chance modifier

combining those two formulae:

P=1-(1-(Cu*(1+M)))n

where:
P is the true DoT chance per shot
Cu is the unmodified (card) DoT chance
M is the DoT chance modifier
n is the number of damage instances (splash and/or pellet count)

Unfortunately, both Flicker and More Pep both have some what low DoT chance modifier values, however, that can be partially rectified with a weapon with a high quantity of instances of damage.

However, while DoT chance cannot hit 100% from multiple procs of <100% chance, a high weapon with a high proc chance can be increased to 100% via modifiers, Fuel the Fire especially.

At this point, you may be wondering: “but what about fire rate? The slagga is also a good slag weapon, but it only has ~50% DoT chance per shot compared to the pimpernel’s 98%, however, the slagga has a much higher FR.”

Well, as fire rate is extreemly obnoxious to model, with many confounding factors (not the least of which is it being tied to frame rate), I’ve elected to model it in terms of the number of shots fired to reach 95% DoT chance instead.

There are 2 ways to model this in terms of shots fired:

T=1-(1-(Cu*(1+M)))(n+s)

where:
T is the target chance
Cu is the unmodified (card) DoT chance
M is the DoT chance modifier
n is the number of damage instances (splash and/or pellet count)
s is the number of shots fired

or

T=1-(1-(1-(1-(Cu*(1+M)))n)s)

where:
T is the target chance
Cu is the unmodified (card) DoT chance
M is the DoT chance modifier
n is the number of damage instances (splash and/or pellet count)
s is the number of shots fired

While the former is easier to read, the later is both easier to solve, and easier to put through a step by step calculation as in the spreadsheet.

This spreadsheet uses 95% or greater as “gaurenteed” DoT chance and provides several comparison metrics, I tried to set it up in such a way that it is mostly independant of fire rate, mag size, and reload speed. Additionaly, it assumes that all pellets connect with the target.

For the time being, I have included only some of the common slag weapons (as I feel this is most applicable to them), if there’s anything I missed / anything you feel should be on here, let me know and I’ll add it.

##Analysis:

##Assumptions:
As with any analysis, I’m making a number of assumptions here and I think it is important to note them, to my knowledge none of these* have been tested, but I feel pretty secure in them because they are either A: very basic, or B: supported by a large body of anecdotal evidence.

• DoT chance is based on the card value
• DoT chance increasing skills multiply with the card chance (* I’m aware that there was probably some testing for this a while back, but I have nothing to link to, so for now I’m leaving it as is)
• all damage instances from a given (DoT capable) weapon, are able to proc a DoT effect
• all damage instances from a given weapon have the same DoT chance
• DoT effects cannot stack
• DoT chance cannot exceed 100%
• DoT effects cannot proc further DoT effects (excluding Electrical Burn and Flame Flare)

##(very abbreviated) Formula derivation:

For the purposes of this explanation, let us assume the the DoT chance of our weapon is 50%, we can freely control the number of damage instances that hit our target, and that the damage procs occur consecutively.

On average, a given damage instance will cause a DoT C% of the time, making C% of the next instance’s hits irrelevant (remember that dot’s don’t stack) resulting in the following pattern:
http://imgur.com/DhTDVH6

Following this method, trying to calculate the true chance with “n” instances of damage gives us this infinite series:

Needless to say this quite a bit of a hassle to calculate.

An interesting pattern arises if we examine the chance to not apply a DoT (1-C):
http://imgur.com/yyLUmXV

Using this pattern we can determine this formula:

B=F n

where:
B is the true fail chance
F is the listed (card) fail chance (1-C)
n is the number of damage instances (splash and/or pellet count)

manipulating this formula to use and result in the correct values:

P=1-(1-C)n

where:
P is the true DoT chance per shot
C is the listed (card) DoT chance
n is the number of damage instances (splash and/or pellet count)

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I’ve updated this, hopefully it’s a bit more useful now, I’m not done with the spreadsheet, but I’ll get that up as soon as I can and update this comment when it’s up.
The spreadsheet is up, as I mentioned in the OP, I just put up some common slag weapons, if I missed anything, or there is anything else that you think should be in there, let me know (I’m looking at you hellborn krieg and shock gaige players). I espically need shock weapons for gaige, becasue I wasn’t sure what to put there, I’ll be checking out chuck’s DoT damage guide in the morning, but for now, I’m tired as fu.

As usual, thank you in advance for helping me fix my \$^@%.

Side note:

forums Forums that are awesome forums that support GFM tables forums that are super-extra-bonus-awesome
the gearbox forums check - -
forums that support GFM tables check check -
other - - -
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