Stacking advantage: doing the math

[howandwhy99]

Stacking advantages would make failure highly unlikely, especially if the target numbers stick in the sweet spot. I think we'll likely see a broader sweet spot because of them though. Say 10 wide?

Bog standard D&D is about increasing one's odds of success or failure before purposely putting the character in a position to metaphorically (and the player literally to) roll the dice. That is if you have to roll at all or something doesn't foul up and you end up rolling early.

Getting Advantage makes this 1-stop shopping, so it isn't exactly appealing to me. However, I think there will be more game to the game, more complex options to add bonuses to oneself and penalties to the other guy, in the later modular add-ons. As a core game it's pretty simple.

 

[Ainamacar]

In a post in my thread about a tiered skill system I made a few charts to demonstrate how multiple dice work out, with I copy here with minor changes. (Some of you may be interested to check out that thread, since characters with the lowest tier of training performing tasks of the lowest tier of difficulty in that system work very similarly to advantage/disadvantage. It's not quite a superset of the a/d system, but it's close.)

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The math of multiple dice is pretty straightforward. If p is the probability of success on a single roll, then the probability of getting at least one success over n rolls is 1-(1-p)^n. Here's a couple tables illustrating that. The first one is just the probability of getting at least one success. The second table is the increase in probability compared to a single roll, which is just 1-(1-p)^n-p.

As you can see from the second graph, a single extra die gives anywhere makes it anywhere up to 25% more likely to get at least one success. On average it is just under 16%, roughly a +3 bonus, assuming all the p occur equally often during play. Probably checks in the middle of the range are more common in play, so the actual benefit of a single extra die is probably a +4 or so. It is worth keeping in mind, however, that a reroll isn't that helpful on very easy or very difficult checks, and that it can't make a check an automatic success or prevent an automatic failure like a +4 potentially could.

The case with rolling 3 dice is very similar, increasing the probability of getting at least one success up to just under 40%. On average (again assuming all probabilities occur with equal frequency in play) the increase is about 24%, about a +5. Again, rolls for p somewhere in the middle are probably more common in play, so the actual improvement is probably closer to a +6.

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If one wanted to stack disadvantage the math is even simpler, because using the worst die means all the dice rolled must succeed. That would make the chance of success p^n. Even for large p (but still less than one) this decays rapidly.

 

[Kinak]

So if I understand this correctly, then if I'm rolling to attack and I have advantage, then I have a 9.75% chance of rolling a critical hit instead of the standard 5% chance?

You are correct.

Tying that to the stacking advantage, that would give you 14.26% on double advantage, 18.54% on triple advantage, and 22.62% on quadruple advantage.

Also, crits just basically never happen with disadvantage. But, oh, will they be epic when they do.

Cheers!
Kinak

 

[slobster]

Stacking advantage works pretty well, giving you a pretty aggressive case of nonetheless diminishing returns. Others have run the numbers pretty well.

What about stacking disadvantage? To use the baseline success rate of 60%, disadvantage (singular) already reduces your chance to succeed to 36%, since you have to succeed on both rolls! That's essentially a -5 penalty, though it also doubles (basically) your chances of an auto-failure and makes a crit exceedingly unlikely.

Double disadvantage takes your success rate down to 21.6%, an effective -3 penalty. Triple disadvantage takes it down to 13%.

The following table lists the effects of stacking advantage and disadvantage at 4 representative points. These points are given assuming a normal roll; so data here represents the effects of stacking disadvantage on a roll that would normally succeed 10% of the time, 30%, 60%, and 80%. The numbers in parenthesis are the effective flat numerical bonus given by stacking the advantage or disadvantage.

[EDIT] I apologize for the lazy image formatting, I don't know how to add tables in these comment windows. [/EDIT]


One thing to notice here is that advantage is more effective at improving rolls in a sweet spot towards the middle, and is better at improving low rolls than improving good rolls (diminishing returns). The corollary is that disadvantage works the inverse way, and is better at wrecking what should be a good roll than it is at exacerbating an already bad roll.

Food for thought.

 

[Danzauker]

I'm kinda with you there. If you've made the effort and arranged a situation so as to make your task incredibly easy, then it should be incredibly easy. That's certainly how people operate in the real world!

Absolutely!!

One of the philosophies of 5ED seems to be "roll only when it's needed".

With advantage rolling 15 or more becomes an average check (about 50% success). With double advantage 17 becomes the average. With triple 18. There's no point of rolling with more than triple advantage.

If the player can convince me that he has earned triple advantage with roleplaying, skill or intuition, he deserves an automatic success.

 

[am181d]

The math here is super easy, right? For advantage, just calculate the miss chance. For disadvantage, calculate the success chance.

And just multiply the percentage chance for each die:

1 die: 50%
2 die: 25% (50% x 50%)
3 die: 12.5% (50% x 50% x 50%)

So if you were rolling 3 dice for advantage and you had a 50% chance on each die, you'd have an 87.5% success rate. If you had disadvantage at the same odds, you'd have a 12.5% success rate.

 

[Kzach]

Also, crits just basically never happen with disadvantage. But, oh, will they be epic when they do.

Heh, you'd have to roll 2x20, right?

Also, conversely on disadvantage, you'd have a 9.75% chance to roll a fumble, yeah?

See... I *like* this. The Avenger was one of my favourite classes solely on the basis of getting to roll two 20's every turn. The very ESSENCE of D&D is rolling a crit or a fumble at a pivotal point. This just increases the odds of awesome.

 

[Kinak]

Heh, you'd have to roll 2x20, right?

Also, conversely on disadvantage, you'd have a 9.75% chance to roll a fumble, yeah?

You got it

See... I *like* this. The Avenger was one of my favourite classes solely on the basis of getting to roll two 20's every turn. The very ESSENCE of D&D is rolling a crit or a fumble at a pivotal point. This just increases the odds of awesome.

With you 100%. I absolutely love the advantage system and can't wait to see it in play.

Cheers!
Kinak

 

[UngeheuerLich]

So from three dice onwards, we're looking at near certainty? Yeah, that does sound too much. I was hoping the diminishing returns would be more pronounced.

I don´t think it would be too much.

After all, if the rogue has a lot of advantages and a good hit chance to begin with, why should he not have good chances. It is exactly what you want: roleplaying well to make your outcome most certain. Which would encourage the rogue to play his role.

 

[Radiating Gnome]

Suddenly, Obiwan having the high ground in Episode 3 makes a lot more sense -- he had advantage!