Advantage

For a comprehensive summary of the mechanics behind all rolls and modifiers, see: Die Rolls

Advantage and Disadvantage are a gameplay mechanic that can greatly affect the success of dice rolls. They only apply to  d20 rolls: Attack Rolls, Saving Throws, and Ability Checks. They never apply to Damage Rolls, though other features and effects can make a character re-roll damage dice in other ways.

Advantage

When you roll with Advantage, you roll two dice and use the higher result. You either have advantage or you don't: it doesn't stack to grant more than a second die, regardless of how many sources of advantage you have.

Example: You roll two  d20 for an Attack Roll, the results are 16 and 4. Your effective result is 16.

Having advantage raises the average of your roll by 3.325 to 13.825. (For the math, see below.)

Examples of situations that grant Advantage on attack rolls:

• Attacking an enemy that is under these conditions: Restrained, Prone, Sleeping, Entangled, Paralysed, Off balance, Enwebbed, Blinded.
• Attacking an enemy while being Hidden or invisible.
• Armour, Weapons, and Spells that grant advantage when attacking enemies of a specific Race.

Disadvantage

When you roll with Disadvantage, you roll two dice and use the lower result. As with advantage, you either have disadvantage or you don't: it doesn't stack to force you to roll more than a second die, regardless of how many sources of disadvantage you have.

Example: You roll two  d20 for an Attack Roll. The results are 16 and 4. Your effective result is 4.

On a D20, having disadvantage lowers the average of your roll by 3.325 to 7.175. (For the math, see below.)

Examples of situations that grant Disadvantage on attack rolls:

• Trying to make a ranged attack against an enemy that is within 5ft and making you Threatened.
• Various spells and abilities that grant Disadvantage.

Advantage and Disadvantage

Having both Advantage and Disadvantage means they cancel each other out, and you roll one die as if you had neither. Because neither advantage or disadvantage stack, having multiple sources of either doesn't change this: even if you have three sources of Advantage, a single source of Disadvantage will cancel it, and vice versa.

Math

Chances of succeeding a specific roll

The benefits of rolling with advantage (or the detriments of rolling with disadvantage) change depending on the target number you need on the 1d20 roll to succeed. The bonus from advantage can be as large as 24-25% when needing a 9, 10, 11, 12, or 13 on the 1d20 roll, and as small as 9% if one needs to roll a 19.

Chance of rolling a target number or above on 1d20

Target on 1d20	Normal Roll	Roll With Advantage	Roll With Disadvantage
1	100%	100%	100%
2	95%	99.75%	90.25%
3	90%	99%	81%
4	85%	97.75%	72.25%
5	80%	96%	64%
6	75%	93.75%	56.25%
7	70%	91%	49%
8	65%	87.75%	42.25%
9	60%	84%	42.25%
10	55%	79.75%	30.25%
11	50%	75%	25%
12	45%	69.75%	20.25%
13	40%	64%	16%
14	35%	57.75%	12.25%
15	30%	51%	9%
16	25%	43.75%	6.25%
17	20%	36%	4%
18	15%	27.75%	2.25%
19	10%	19%	1%
20	5%	9.75%	0.25%

Effects on the average of dice rolls

A more general way of looking at advantage/disadvantage is calculating the effect on the average of dice rolls. This makes it more broadly applicable than looking at specific rolls and makes it easier to compare to other bonuses and penalties which may apply to a roll.

For this we first need to clarify the notations used below: D$n$ represents an $n$-sided die, $P(i)$ is the probability that a variable has value $a$, $\mathbb {E}$ denotes the average or expected value of a roll, and $\textstyle \sum _{i=a}^{b}x_{i}$ denotes the sum of a series of numbers $x$ over an index $i$ with $i$ going from $a$ through $b$.

The formula to calculate the expected value, $\mathbb {E} [x]$, of a variable $x$ is equal to the sum of every possible value of $x$ multiplied by the chance for $x$ to have that value. In the case of an $n$-sided die, D$n$, this becomes:

$\mathbb {E} [{\text{D}}n]=\sum _{i=1}^{n}(i\cdot P(i))$

For a regular dice roll the probability distribution is uniform, which means $P(i)=1/n$ for any $i$, and using $\sum _{i=1}^{n}i={\frac {1}{2}}n(n+1)$, we get

$\mathbb {E} [{\text{D}}n]=\sum _{i=1}^{n}(i\cdot P(i))={\frac {1}{n}}\left({\frac {n(n+1)}{2}}\right)={\frac {n+1}{2}}$

For a dice roll with advantage the chance to roll the number $i$ is equal to the chance that the first die rolls $i$ multiplied by the chance that the second die rolls $i$ or less, multiplied by 2 (because the 2 dice are interchangeable), minus the chance of both dice rolling $i$ (because we counted that possibility twice by multiplying by 2). This gives

$P_{\text{adv}}(i)=2P(i)\sum _{j=1}^{i}P(j)-P(i)^{2}=2{\frac {1}{n}}\cdot {\frac {i}{n}}-{\frac {1}{n^{2}}}={\frac {2i-1}{n^{2}}}$

Applying that to the formula of an average of a die D$n$ we get

$\mathbb {E} [{\text{D}}n{\text{ with advantage}}]=\sum _{i=1}^{n}i\cdot {\frac {2i-1}{n^{2}}}={\frac {2}{n^{2}}}\cdot \sum _{i=1}^{n}i^{2}-{\frac {1}{n^{2}}}\cdot \sum _{i=1}^{n}i$

Here we can use that the sum of squares is $\sum _{i=1}^{n}i^{2}={\frac {1}{6}}n(n+1)(2n+1)$, which gives

$\mathbb {E} [{\text{D}}n{\text{ with advantage}}]={\frac {2}{n^{2}}}\left({\frac {n(n+1)(2n+1)}{6}}\right)-{\frac {1}{n^{2}}}\left({\frac {n(n+1)}{2}}\right)={\frac {2n}{3}}+1+{\frac {1}{3n}}-{\frac {1}{2}}-{\frac {1}{2n}}={\frac {2n}{3}}+{\frac {1}{2}}-{\frac {1}{6n}}$

To know what bonus having advantage gives to our roll, we calculate

$\mathbb {E} [{\text{D}}n{\text{ with advantage}}]-\mathbb {E} [{\text{D}}n]={\frac {2n}{3}}+{\frac {1}{2}}-{\frac {1}{6n}}-{\frac {n+1}{2}}={\frac {1}{6}}\left(n-{\frac {1}{n}}\right)$

When we apply this elegant expression to a D20 we get that having advantage is equivalent to an average bonus of +3.325.

Because of symmetry, having disadvantage instead of advantage means we can simply make the permutation of $\{1,\dots ,n\}\to \{n,\dots ,1\}$ for the values of dice rolls and all the calculations will remain the same. Therefore the size of the bonus of advantage is equal to the size of the penalty of disadvantage.

Application: Savage Attacker

The Savage Attacker feat essentially gives advantage to melee damage rolls. We can use the result of the calculations above to see what the average bonus to our damage becomes, depending on what dice the weapon uses.

Expected Bonus Damage from Savage Attacker

Weapon Die	Average Bonus Damage
1d4	0.625
1d6	0.972
1d8	1.3125
1d10	1.65
1d12	1.9861
2d6	1.94

Note that Savage Attacker also applies to ALL additional damage dice from ANY source added to a weapon, but not Sneak Damage because those are not bonus dice added to the weapon damage. For example, the Halberd of Vigilance (d10 slashing damage and d4 force damage) which was dipped in fire (d4 fire damage) will, on average, do 1.65 + 0.625 +0.625 = 2.9 more damage with Savage Attacker.

External Links

• The unexpected logic behind rolling multiple dice and picking the highest by Matt Parker
• Advantage and Disadvantage in D&D Next: The Math by The Online Dungeon Master (Michael Iachini)
• D&D 5e: Probabilities for Advantage and Disadvantage by Bob Carpenter