- Gnome Stew - https://gnomestew.com -

Advantage Vs Disadvantage – How the numbers stack up

The new 5th edition of Dungeons and Dragons has tweaked some of it’s systems somewhat. One of them is the new advantage/disadvantage system which replaces the system standard plus or minus 2. In addition, advantage and disadvantage doesn’t stack and any amount of each cancels out the other, so if I have disadvantage already, I might as well pick up as much of it as I can: it won’t hurt anything, and if I have disadvantage from five sources, I can level the playing field by burning a single ability that gives me advantage. This sounds easy, eliminates a lot of the fiddly “did you remember the +2 for this, —1 for this, —2 for this, and +1 for this? Oh yeah and flanking, total bonus is… +2” and is likely to encourage players to pull out all the stops (or allow GMs to pile on ridiculous situations) since they no longer stack.

“I have to shoot an arrow and hit the lever on the other side of the 100 foot wide chasm to lower the bridge? That’s disadvantage because of the range. It’ll be hard. What? I have to do it in a heavy crosswind, hanging from a rope with one hand, nocking my arrow with my teeth while the kilt of the flatulent dwarf dangling over me blocks one of my eyes? Meh. it’s not like it’s any harder that way.” — disadvantage at work

The main problem with this new system, from my perspective, is that it’s not as simple to wrap your brain around what having advantage or disadvantage means as the old system. For example, under the new system, I can make a halfling warrior that swings around a greataxe twice as long as he is… but all my attacks have disadvantage… Is that worth it? I can also use armor I’m not proficient in, but I get disadvantage to all rolls involving strength and dex. Is that worth it? (Short answer: No. Almost never worth it.) I wanted to know for my own benefit, so I did all the heavy lifting and here are the results:

First, what is the probability of rolling any given number on a d20? What are the chances if you add a +2 and what are the chances if you subtract —2? That one’s easy. If you can actually roll the number, the probability of rolling it is 1/20. What adding and subtracting 2 does isn’t change the odds of rolling a particular number, but instead changes the numbers you can roll. I’m not going to graph those odds. It would look like three mostly overlapping straight lines. Not terribly interesting.

What about advantage and disadvantage? First, unlike the ±2 solution, this new mechanic doesn’t change the numbers you can roll, which is nice, only the chances of rolling them. With advantage, the probability of rolling a 1 is a lot lower than usual .05. It’s .0025, one twentieth of normal. The probability of rolling 20, on the other hand is almost double the usual .05. It’s .0975. With disadvantage, these numbers are reversed. 20’s are almost impossible, 1s are twice as likely as usual. This one I will graph:

P(roll) [1]

The exact formula for the probability of getting a roll (x) with advantage is:
P([1st roll is x AND 2nd is less] OR [2nd is x AND 1st is less] OR [1st is x AND 2nd is x])
=[1/20 * (x-1)/20] + [(x-1)/20 * 1/20] + [1/20 * 1/20]
=(x-1)/400 + (x-1)/400 + 1/400
=2(x-1)/400 + 1/400
=(2x-2)/400 +1/400
=(2x — 1)/400
=(1/200)x —(1/400) which is of the form y=mx+b which explains why we have nice straight lines

With disadvantage, the math is very similar (I leave most of it to the reader) but the (x-1) above is replaced by (20-x) so the end result is: -(1/200)x +41/400.

But we rarely have to roll an exact number, instead we need to beat a DC, so the probability of that is the sum of the probability of rolling that number plus the probability of rolling every higher number. You can think of this as (kind of) the area under the lines on the chart above to the right of the DC of interest. Thus you can see that under that standard system, as DC increases, you lose the same amount of probability of success for each step. With advantage, however, you lose much smaller amounts of probability at first, but these amounts get larger and larger over time. This means that while the probability of success of a straight die roll or a +2 adjustment for different DCs is a straight line, for advantage, it’s a parabolic curve (wolfram alpha says it’s almost exactly 1 — X^2/400)P(Success Advantage) [2]You can see from this graph that the probability of success from adding 2 to your roll is always the same 10% bonus, but the bonus granted by advantage isn’t static. It’s highest in the middle and lowest on the extremes. vs a DC 11, it’s a 25% bonus, the same as a +5 on your roll. At DC2 and DC20 advantage is almost equivalent to a +1 bonus (+.0475 vs +.05). on average, the bonus advantage grants is +.16625, a little over a +3 bonus (+.15). From this chart it’s hard to argue with the conclusion that advantage is better mechanically than the +2 bonus, but keep in mind that the area where advantage falls down is precisely where you need it. It adds less probability of success than the +2 bonus for rolls vs DC 19 (+.09) and 20 (+.0475). Where advantage truly shines isn’t in moving a poor chance to a reasonable chance, but rather turning a 50-50 chance into a good chance.

Disadvantage’s success chance graph looks similar, but mirrored:

P(Success Disadvantage) [3] Everything that we said about advantage also applies to disadvantage. It’s strongest in the mid ranges and weakest at the extremes. DC11 has a —.25 penalty, same as a —5 on your roll, while for DC2 and DC20 it’s almost equivalent to a —1 (-.0475). On average, disadvantage is a little worse than a —3 (-.16625). It’s worse than a —2 bonus on DCs 4-18 but better than a —2 bonus at DCs 2, 3, 19 and 20. Again, disadvantage doesn’t make a huge difference at the extreme DCs but it’s very good at turning a 50-50 chance into a poor chance.

So taken as a whole, the advantage/disadvantage system has little impact on extreme situations but can have a large impact on middle of the road situations and is on average larger in impact that the system in use in earlier systems, so work to get advantage and avoid disadvantage when you can, even more so than prior editions. This change may have some interesting implications for common play styles and the overall mathematical systems of the game.

P.S. If you’d like to see the spreadsheet I used to make those graphs (plus a bonus graph!) here’s a link to it [4].

20 Comments (Open | Close)

20 Comments To "Advantage Vs Disadvantage – How the numbers stack up"

#1 Comment By bislab On September 10, 2014 @ 3:12 am

In your opening description, I’d like to add that while piling on the difficulty to the situation doesn’t increase the amount of disadvantage dice…it WILL increase the DC of the action you are trying to accomplish. So your archer hanging by a thread in the wind making the impossible shot may ONLY have disadvantage, but likely his DC is 20.

#2 Comment By Matthew J. Neagley On September 10, 2014 @ 4:24 am

I’ve only got the quick start rules so the full ones may have more/better clarification, but I don’t see anything about changing the DC from complications, only adding disadvantage.

#3 Comment By Airk On September 10, 2014 @ 9:39 am

While I haven’t read the rules, I would be super disappointed to discover that the DM can just crank up the target number to reflect increased difficulty. That’s literally exactly the same as assigning penalties, and would pretty much defeat the point of the whole advantage/disadvantage system.

#4 Comment By Matthew J. Neagley On September 10, 2014 @ 3:40 pm

I wouldn’t be surprised if there was a provision for sliding the DC about, but like you I can’t say I would like it mostly because of a lack of consistency. Which things add advantage/disadvantage and which shift the DC? When you stack them where is the tipping point where you say “This much was all just disadvantage, but one more penalty and I start moving the DC.”

It starts to break the simple elegance of the system.

#5 Comment By bislab On September 11, 2014 @ 2:27 am

What you are describing would be an Ability/skill check, probably a DEX(acrobatics) roll. DC’s are assigned a class of (5, 10, 15, 20, 25, 30) as per pg 142 of the Players handbook. So you get Advantage/Disadvantage on the ability check and if you are trained in the skill of Acrobatics you get to add in your Proficiency bonus and your Ability bonus if you have one. So …. yeah, it is officially there.

#6 Comment By 77IM On September 11, 2014 @ 9:41 am

If there are multiple hazardous circumstances at work I think, by the rules, the DM can just impose more checks. For example, a Strength (Athletics) check to hang upside-down, a Dexterity (Acrobatics) check to not sway too much in the wind, etc.

Aside from such trickery I think the intention of 5E is that the DM should waive checks when the situation seems too easy or too impossible. (On page 58 of Player’s Basic Rules, “When the outcome is uncertain, the dice determine the results.”) If I were the DM and a character were facing 5 disadvantages I’d probably just rule that the outcome is certain and they fail.

#7 Comment By Rickard Elimää On September 10, 2014 @ 4:04 am

I rely on Anydice when I want to see any probabilities. I did that to see how much bonus the Avenger got in D&D4 with that extra d20, and it’s basically a +5 bonus. (Click graph in link below to see comparisons.)


What could be noticed is also that advantage doubles the chance to crit. Personally, I like that side effect.

But in the end, I don’t see the point in comparing +2 bonus to the advantage system. That gives the impression that a +2 bonus is how it should work.

#8 Comment By Matthew J. Neagley On September 10, 2014 @ 4:39 am

I’m comparing the two, not because +2/-2 is the “way it should be” but because it’s the way we’re used to and understand so it makes a handy departure point for better understanding of advantage disadvantage.

Also, take another look at those charts (yours or mine, they’re the same) that +5 bonus is the maximum increase (found at DC 11) everywhere else, the bonus is weaker.

#9 Comment By IvanXXIII On September 10, 2014 @ 9:23 am

Advantage does not quite double the chance to crit. The formula for rolling a critical hit (assuming the standard 5% range) is the chance to roll a critical on the first dice plus the chance to roll a critical on the second minus the chance to roll a critical on both dice. Basically, you have to subtract the chance that you roll two 20’s because when adding the probability of both dice you will double count the 20/20 roll.

In mathematical notation:
P(A U B) = P(A) + P(B) – P(AB)
= 0.05 + 0.05 – 0.0025
= .0975

Where P(A) is the probability of the first die to roll a critical (0.05). P(B) is the probability of the second die to roll a critical (0.05). P(AB) is the probability to roll a critical on both dice at the same time (0.05*0.05=.0975). A U B is the union of A and B or the set containing the elements from both A and B and P(A U B) is the probability of rolling a critical on either A or B. (Note: here “or” is inclusive i.e. A, B, and A&B)

P.S. I apologize if you were rounding and I didn’t need to correct you.

P.P.S. Notice a fighter with a range of 18-20 will critical 27.75% of the time. This is left as an exercise for the reader. 😛

#10 Comment By Rickard Elimää On September 10, 2014 @ 2:30 pm

Of course I was rounding up. The result in Anydice showed that.

#11 Comment By Ayronis On September 10, 2014 @ 8:22 am

Thank you for this post. I have been dying to hear someone with experience in both systems talk about this feature in some detail. This was exactly the kind of investigation I was hoping for. The differences in the mid-range could have a serious impact on mid-level game play. Very curious to see how this plays out in the coming years.

#12 Comment By Airk On September 10, 2014 @ 9:38 am

Careful there. The “mid range” doesn’t mean “mid level play” it just means “when you need a middle-of-the-road number” (as opposed to “I only need to roll 4 or higher” or “I need a natural 20.” or something.)

#13 Comment By encoded On September 10, 2014 @ 9:59 am

[K]eep in mind that the area where advantage falls down is precisely where you need it. It adds less probability of success than the +2 bonus for rolls vs DC 19 (+.09) and 20 (+.0475).

While it’s true that the additional probability is smaller, the % improvement in your chances is greater. That is for a DC 20, your chances of success are (nearly) doubled, for an improvement of 100%. For a DC 10, your chances of success improve by only 50%.

In any event, you nailed it when you wrote:

The main problem with this new system, from my perspective, is that it’s not as simple to wrap your brain around what having advantage or disadvantage means as the old system.

#14 Comment By Matthew J. Neagley On September 10, 2014 @ 4:12 pm

You’re not the only person to address the percent improvement for critical hits being so large, and you’re right you do (nearly) improve your chance to crit by 100%.

However, in general I try to stay away from percent improvement as a metric because it tends to misrepresent the magnitude of the effect, in many cases innocently but in many more cases deliberately.

Instead I like to stick with absolute improvements (in this case about 5%) because they are easier to understand mechanically and harder to abuse.

In this case, saying the effect of advantage is about a +5% chance to hit/crit or double the chance to crit are both easy to understand and the magnitudes are fairly clear, but the smaller the magnitude of your original effect, the easier it is for an improvement of no practical significance to be passed off as a huge percent improvement.

You can see this in a lot of nonsense statistics (and it featured quite embarrassingly in a landmark lawsuit in Italy a few years ago) where they will say for example that increasing from a one in a billion chance of something happening to a one in 10 million is 100 times greater risk (oh no! How horrible! It’s going to happen just any day now! Donate to our cause! Sign our petition! Change your entire lifestyle! Class action lawsuit!)

Further, it’s really easy to manufacture massive percent improvements where there are none. If, for example a success rate changes from 99% to 97%, that’s the same as the failure rate changing from 1% to 3% Uh-oh! That’s a 200% increase in failure rate! What a massive difference!

It’s pure nonsense, and abuses like this make percent improvement one of those statistics that you should Always, always, always look at with a healthy dose of incredulity.

#15 Comment By David Russell On September 10, 2014 @ 3:14 pm

There are still modifiers, but its up to the DM to use them. the most common modifier is COVER, cover is either +2 to the DC (or AC) if its half, +5 to the DC (or ac if its 3/4) So if one of your eyes is being blocked it would give the targets AC a +2.

And it specifically states in the rules that the Dm can give modifiers to emphasize other factoring conditions. they SUGGEST +/- 2 for small conditions and +/- 5 for greater conditions. the difference is that the conditions other than Cover are not DEFINED but instead left up to the DM to decide. (this edition leaves a lot more to the DM than last 2 editions have done) so if i was your DM and you where trying to shoot a level on the other side of the Room I would give your attack a -5 for difficulty, and the target would get +2 to DC for cover (unless you were able to adjust your head and not have your vision impaired.

#16 Comment By Matthew J. Neagley On September 10, 2014 @ 3:55 pm

Wouldn’t having one eye covered give your target a 1/2 cover bonus imply that if I close one eye I can only see half of the target? :p

#17 Comment By Chris Waldrip On September 10, 2014 @ 9:57 pm

I haven’t had to do it yet, but I’ve thought about “Double-Dis/Advantage”. The example given above, or maybe a Human Ranger trying to shoot at a exposed spot of a flying dragon’s belly at long range, would be examples. of making the player roll 3d20, and take the lowest or highest.

I wonder what that does to the odds…

#18 Comment By 77IM On September 11, 2014 @ 9:52 am

This is a really good analysis. Two points that people may be missing:

1. Because of 5E’s “bounded accuracy,” the mid-range DCs are going to be MUCH more common. Those are the DCs where the probability change is very close to +/-5. The ends of the curve where the probability change rapidly tapers down should be encountered very rarely in game play.

2. For attack rolls, because a natural 20 is a crit and deals extra damage, you can apply the expected value of that damage and advantage/disadvantage can become worth slightly MORE that +/-5.

Overall I think this means it’s safe to treat advantage/disadvantage as a flat +/-5. (By “safe” I mean you will make the right decisions if you think of it this way.)

#19 Comment By Kurt “Telas” Schneider On September 18, 2014 @ 8:23 am

I’m back to gaming, running a Dwarf Cleric in 5E. I like this system so far. Take all the best parts of the previous four versions, buff out all the rough edges, and add some mandatory RP elements (background, Inspiration).

Thank you for this! I was thinking of emailing you for a professional opinion when I first started seeing the discussions of ad/disad.

One thing to remember is that 5E has a less open-ended system for DCs than 3.x did. There is basically a soft upper limit to DCs, and the advancement rate shows it. In my limited experience (7th level one-shot, three campaign sessions, experienced GM), the vast majority of DCs are in the middle of the characters’ range.

#20 Comment By R H Manifold On February 4, 2018 @ 9:13 am

There are mechanics that activate on a spread (19-20, frx) the probability of which would be affected by advantage/disadvantage. So it’s not just a matter of accuracy, but of hitting specific target numbers on the dice. Advantage indeed.