The math of Advantage and Disadvantage

[Keravath]

There seems to be a lot of misinformation about the effects of advantage, disadvantage and elven accuracy (or the lucky feat) on the chances to hit, skill checks and saving throws. Some folks seem to consider them equivalent to +/-3 ... others +/-5. However, the sources some of them seem to reference are either incorrect or incomplete.

So in case anyone is interested ... the math of advantage


----

Just for interest sake ... here is the math for advantage and disadvantage as a function of the number you need to roll to hit.

If the number you need to roll to hit is X ... e.g. X + to hit modifier = AC then the formulae are (X has a minimum value of 2 since a 1 always misses and a maximum value of 20 since 20 always hits).

1) Advantage

Advantage probability to hit is: 1.0 - the odds that both die rolls miss.

Advantage % to hit X = 100 * ( 1 - [(X-1)/20]^2)

e.g. AC=15, to hit modifier is +5, to hit = 10
Advantage % to hit = 79.8%

2) Disadvantage

Disadvantage probability to hit is: probability that both die rolls hit

Disadvantage % to hit X = 100 * [(20-X+1)/20]^2

e.g. AC=15, to hit modifier is +5, to hit = 10
Disadvantage % to hit = 30.6%

3) Tri-vantage (elven accuracy or lucky feat applied to advantage)

Tri-vantage probability to hit is: 1.0 - the odds that all three die rolls miss.

Tri-advantage % to hit X = 100 * ( 1 - [(X-1)/20]^3)

e.g. AC=15, to hit modifier is +5, to hit = 10
Tri-vantage % to hit = 90.9%

---------------

The base probability to get a 10 or greater result without advantage or disadvantage is 55%

Thus in this case, disadvantage reduces the to hit by about 25% while advantage increases it by about 25%.

THIS is why passive skills are modified by +/-5 for advantage and disadvantage since for the average case advantage/disadvantage incur a +/-25% success probability change which is roughly equivalent to a static +/-5.

----------------

So I just thought I would add a table showing the % to hits for normal/advantage/disadvantage/trivantage vs target number along with the difference from the base case.

[TABLE="class: cms_table"]
[TR]
[TD]X[/TD]
[TD]Normal %[/TD]
[TD]Advantage[/TD]
[TD]% Diff[/TD]
[TD]Disadvantage[/TD]
[TD]% Diff[/TD]
[TD]Tri-vantage[/TD]
[TD]%Diff Base[/TD]
[TD]%Diff Adv[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]100[/TD]
[TD]100[/TD]
[TD]0[/TD]
[TD]100[/TD]
[TD]0[/TD]
[TD]100[/TD]
[TD]0[/TD]
[TD]0[/TD]
[/TR]
[TR]
[TD]2[/TD]
[TD]95[/TD]
[TD]99.75[/TD]
[TD]4.75[/TD]
[TD]90.25[/TD]
[TD]-4.75[/TD]
[TD]99.99[/TD]
[TD]5.0[/TD]
[TD]0.2[/TD]
[/TR]
[TR]
[TD]3[/TD]
[TD]90[/TD]
[TD]99[/TD]
[TD]9[/TD]
[TD]81[/TD]
[TD]-9[/TD]
[TD]99.9[/TD]
[TD]9.9[/TD]
[TD]0.9[/TD]
[/TR]
[TR]
[TD]4[/TD]
[TD]85[/TD]
[TD]97.75[/TD]
[TD]12.75[/TD]
[TD]72.75[/TD]
[TD]-12.75[/TD]
[TD]99.7
[/TD]
[TD]14.7[/TD]
[TD]1.9[/TD]
[/TR]
[TR]
[TD]5[/TD]
[TD]80[/TD]
[TD]96[/TD]
[TD]16[/TD]
[TD]64[/TD]
[TD]-16[/TD]
[TD]99.2
[/TD]
[TD]19.2[/TD]
[TD]3.2[/TD]
[/TR]
[TR]
[TD]6[/TD]
[TD]75[/TD]
[TD]93.75[/TD]
[TD]18.75[/TD]
[TD]56.25[/TD]
[TD]-18.75[/TD]
[TD]98.4
[/TD]
[TD]23.4[/TD]
[TD]4.7[/TD]
[/TR]
[TR]
[TD]7[/TD]
[TD]70[/TD]
[TD]91[/TD]
[TD]21[/TD]
[TD]49[/TD]
[TD]-21[/TD]
[TD]97.3
[/TD]
[TD]27.3[/TD]
[TD]6.3[/TD]
[/TR]
[TR]
[TD]8[/TD]
[TD]65[/TD]
[TD]87.75[/TD]
[TD]22.75[/TD]
[TD]42.25[/TD]
[TD]-22.75[/TD]
[TD]95.7
[/TD]
[TD]30.7[/TD]
[TD]8.0[/TD]
[/TR]
[TR]
[TD]9[/TD]
[TD]60[/TD]
[TD]84[/TD]
[TD]24[/TD]
[TD]36[/TD]
[TD]-24[/TD]
[TD]93.6
[/TD]
[TD]33.6[/TD]
[TD]9.6[/TD]
[/TR]
[TR]
[TD]10[/TD]
[TD]55[/TD]
[TD]79.75[/TD]
[TD]24.75[/TD]
[TD]30.25[/TD]
[TD]-24.75[/TD]
[TD]90.9
[/TD]
[TD]35.9[/TD]
[TD]11.1[/TD]
[/TR]
[TR]
[TD]11[/TD]
[TD]50[/TD]
[TD]75[/TD]
[TD]25[/TD]
[TD]25[/TD]
[TD]-25[/TD]
[TD]87.5
[/TD]
[TD]37.5[/TD]
[TD]12.5[/TD]
[/TR]
[TR]
[TD]12[/TD]
[TD]45[/TD]
[TD]69.75[/TD]
[TD]24.75[/TD]
[TD]20.25[/TD]
[TD]-24.75[/TD]
[TD]83.4
[/TD]
[TD]38.4[/TD]
[TD]13.6[/TD]
[/TR]
[TR]
[TD]13[/TD]
[TD]40[/TD]
[TD]64[/TD]
[TD]24[/TD]
[TD]16[/TD]
[TD]-24[/TD]
[TD]78.4
[/TD]
[TD]38.4[/TD]
[TD]14.4[/TD]
[/TR]
[TR]
[TD]14[/TD]
[TD]35[/TD]
[TD]57.75[/TD]
[TD]22.75[/TD]
[TD]12.25[/TD]
[TD]-22.75[/TD]
[TD]72.5
[/TD]
[TD]37.5[/TD]
[TD]14.8[/TD]
[/TR]
[TR]
[TD]15[/TD]
[TD]30[/TD]
[TD]51[/TD]
[TD]21[/TD]
[TD]9[/TD]
[TD]-21[/TD]
[TD]65.7
[/TD]
[TD]35.7[/TD]
[TD]14.7[/TD]
[/TR]
[TR]
[TD]16[/TD]
[TD]25[/TD]
[TD]43.75[/TD]
[TD]18.75[/TD]
[TD]6.25[/TD]
[TD]-18.75[/TD]
[TD]57.8
[/TD]
[TD]32.8[/TD]
[TD]14.1[/TD]
[/TR]
[TR]
[TD]17[/TD]
[TD]20[/TD]
[TD]36[/TD]
[TD]16[/TD]
[TD]4[/TD]
[TD]-16[/TD]
[TD]48.8
[/TD]
[TD]28.8[/TD]
[TD]12.8[/TD]
[/TR]
[TR]
[TD]18[/TD]
[TD]15[/TD]
[TD]27.75[/TD]
[TD]12.75[/TD]
[TD]2.25[/TD]
[TD]-12.75[/TD]
[TD]38.6
[/TD]
[TD]23.6[/TD]
[TD]10.8[/TD]
[/TR]
[TR]
[TD]19[/TD]
[TD]10[/TD]
[TD]19[/TD]
[TD]9[/TD]
[TD]1[/TD]
[TD]-9[/TD]
[TD]27.1
[/TD]
[TD]17.1[/TD]
[TD]8.1[/TD]
[/TR]
[TR]
[TD]20[/TD]
[TD]5[/TD]
[TD]9.75[/TD]
[TD]4.75[/TD]
[TD]0.25[/TD]
[TD]-4.75[/TD]
[TD]14.3
[/TD]
[TD]9.3[/TD]
[TD]4.5[/TD]
[/TR]
[/TABLE]



Notes:

This shows that over the range of the most common target numbers from about 8 to 14 (due to bounded accuracy) the effect of advantage/disadvantage varies from +/-21% to +/-25% (+4 to +5).

At the very extreme of the target numbers ... like needing a natural 20 to hit ... the effect is closer to that of a +1. However, the extremes do not come up as often as the middle of the distribution ... the game is balanced around typical target numbers in a standard encounter around 11. AC16 with +5 to hit at level 3 or maybe a typical AC20 with +9 to hit at level 11 ... sometimes the AC's are much easier or much harder to hit but then the creatures likely have varied hit points or other compensating abilities (like resistances).

Due to this, ascribing a static +/-3 to advantage/disadvantage isn't an accurate assessment.

The effect of elven accuracy "trivantage" is also interesting since for target numbers between 12 and 17 it is the equivalent of +2 to almost +3 to hit compared to regular advantage.

Finally, by comparing the target numbers for a base hit against target numbers with a +5 applied for advantage and trivantage you can assess the impact of using feats like GWM and SS. (eg. if the target number is normally 10 ... it will be 15 when using SS/GWM)


P.S. Keep in mind that the 1 line is only for skill checks (and saving throws?) ... to hit rolls auto miss on a 1 so it uses the "2" line in the table.

 

[Swarmkeeper]

Good stuff!

Thanks!

 

[Charlaquin]

I think the most important thing to take away here is that the impact of (Dis)Advantage varies depend on your target number, and that it has the greatest impact at 11. The further from 11 your target number is, in either direction, the less impact (Dis)Advantage has. Eleven Accuracy has a lower impact than Advantage at all target numbers, but it likewise has the biggest impact at 11 which deminishes the further the target number is. It’s also worth keeping in mind that 5e, by and large, is tuned to keep the target number for most tasks around 8 for trained characters, up to around 14 for untrained characters. There’s obviously some variation, but that’s the range the game’s math aims for. At those points, (Dis)Advantage is a 22.75% difference, which is a little better than +/-4 but a little worse than +/-5. Elven Accuracy gives an additional 8% compared to disadvantage, which is a little better than +1 and a little worse than +2.

 

[Keravath]

I think the most important thing to take away here is that the impact of (Dis)Advantage varies depend on your target number, and that it has the greatest impact at 11. The further from 11 your target number is, in either direction, the less impact (Dis)Advantage has. Eleven Accuracy has a lower impact than Advantage at all target numbers, but it likewise has the biggest impact at 11 which deminishes the further the target number is. It’s also worth keeping in mind that 5e, by and large, is tuned to keep the target number for most tasks around 8 for trained characters, up to around 14 for untrained characters. There’s obviously some variation, but that’s the range the game’s math aims for. At those points, (Dis)Advantage is about a 22% difference, which is a little better than +/-4 but a little worse than +/-5.

Actually, elven accuracy doesn't have its greatest impact at 11 ... elven accuracy has a peak impact at 13 compared to the base target numbers and at 14 when compared to advantage (because advantage starts to fall off faster than trivantage). Elven accuracy has a bigger impact when trying to hit more difficult targets. Elven accuracy by itself is close to an additional +3 over the range of target numbers from 12 to 16.

Over the 8 to 14 target number range mentioned the equivalent modifier is from +4.5 to +5.

 

[TarionzCousin]

I am not great with math. But I do know that two is better than one... unless it's worse than one.

 

[Charlaquin]

Actually, elven accuracy doesn't have its greatest impact at 11 ... elven accuracy has a peak impact at 13 compared to the base target numbers and at 14 when compared to advantage (because advantage starts to fall off faster than trivantage). Elven accuracy has a bigger impact when trying to hit more difficult targets. Elven accuracy by itself is close to an additional +3 over the range of target numbers from 12 to 16.

Over the 8 to 14 target number range mentioned the equivalent modifier is from +4.5 to +5.

Hm. So it is. My mistake.

 

[pming]

Hiya!

Interesting...hmmm....interesting indeed...

...so, what you're saying is: "Advantage/Disadvantage mechanic is fun".

Yup, math checks out.



^_^

Paul L. Ming

 

[Yaarel]

The benefits of advantage were heavily discusses when 5e playtests first announced the concept.

The problem is, the straightforward math fails to take into account the following consideration.

Advantage is beneficial when you dont need it − when the AC is already low.

Advantage is unhelpful when you really need it − when the AC is too high.



When all is said-and-done. Advantage is equal to about +2½ bonus. In other words, if you have a choice between +2 and advantage, take the advantage. When you have a choice between advantage and +3, take the +3.

Flat bonuses of +4 and +5 are much more useful than advantage.

 

[Keravath]

The benefits of advantage were heavily discusses when 5e playtests first announced the concept.

The problem is, the straightforward math fails to take into account the following consideration.

Advantage is beneficial when you dont need it − when the AC is already low.

Advantage is unhelpful when you really need it − when the AC is too high.



When all is said-and-done. Advantage is equal to about +2½ bonus. In other words, if you have a choice between +2 and advantage, take the advantage. When you have a choice between advantage and +3, take the +3.

Flat bonuses of +4 and +5 are much more useful than advantage.

Unfortunately, your comments are incorrect which is why I posted the table in the first place.

Would you agree that an AC20 would be a high AC for a level 1-4 character? A level 1-4 character typically has +5 to hit (+2 proficiency and +3 stat) resulting a 15+ being required to hit. If you READ the table posted above. The effect of advantage when the target number is 15 is an increase of 21% in the chances to hit which is equivalent to just more than a static +4.

To get down to an effective +2.5 you would need to be fighting a creature with an AC of 23 at level 1-4 ... giving a required roll to hit of 18.

Over a wide range of AC (in fact almost anything not considered ridiculously high i.e. AC23 at tier 1 ... AC27 at tier 2-3, +5 stat and +4-5 proficiency), advantage is indeed equivalent to a static +4 to +5. THIS is exactly why I posted both the math and the tables since folks seem to have this mistaken impression that the effect of advantage is about the same as +2.5 to +3 which is generally incorrect.

Flat bonuses of +4 to +5, in most cases, are the equivalent of advantage. The exception being extremely high to hit numbers which come into play only for EXTREMELY high AC. If you have +10 to hit and the AC of the creature you are attacking is 25 (like a Tarrasque?) then advantage is STILL better than than a static +4.

Finally, if you can somehow arrange to get a static +4 on to hit through magical weapons (e.g. +2 bow +2 archery) then advantage ON TOP of this will be the equivalent of an additional +4 to +5.

 

[Yaarel]

Unfortunately, your comments are incorrect which is why I posted the table in the first place.

Would you agree that an AC20 would be a high AC for a level 1-4 character? A level 1-4 character typically has +5 to hit (+2 proficiency and +3 stat) resulting a 15+ being required to hit. If you READ the table posted above. The effect of advantage when the target number is 15 is an increase of 21% in the chances to hit which is equivalent to just more than a static +4.

To get down to an effective +2.5 you would need to be fighting a creature with an AC of 23 at level 1-4 ... giving a required roll to hit of 18.

Over a wide range of AC (in fact almost anything not considered ridiculously high i.e. AC23 at tier 1 ... AC27 at tier 2-3, +5 stat and +4-5 proficiency), advantage is indeed equivalent to a static +4 to +5. THIS is exactly why I posted both the math and the tables since folks seem to have this mistaken impression that the effect of advantage is about the same as +2.5 to +3 which is generally incorrect.

Flat bonuses of +4 to +5, in most cases, are the equivalent of advantage. The exception being extremely high to hit numbers which come into play only for EXTREMELY high AC. If you have +10 to hit and the AC of the creature you are attacking is 25 (like a Tarrasque?) then advantage is STILL better than than a static +4.

Finally, if you can somehow arrange to get a static +4 on to hit through magical weapons (e.g. +2 bow +2 archery) then advantage ON TOP of this will be the equivalent of an additional +4 to +5.

In my experience, what you are saying is true enough for the lowest levels of the ‘apprentice tier’ when the DM tends to coddle the players.

But at higher levels − for more ‘verisimilitudinous’ play styles − heroes are likely to come across creatures whose levels diverge significantly from their own. The AC can vary significantly. Often enough, the advantage proves unhelpful.

To get down to an effective +2.5 you would need to be fighting a creature with an AC of 23 at level 1-4 ... giving a required roll to hit of 18.

Yes, correct. The need for a natural 18 or higher happens often enough (especially for skill checks), and the advantage mechanic becomes less helpful when one truly needs a bonus.