Editing Advantage

{{NoExcerpt|''For a comprehensive summary of the mechanics behind all rolls and modifiers, see: [[Dice Rolls]]''}}
{{Advantage}} and {{Disadvantage}} are gameplay mechanics that can greatly affect the success of dice rolls.  They only apply to {{D20}} rolls: [[Attack Roll]]s, [[Saving Throw]]s, and [[Ability Check]]s. They never apply to [[Damage Roll]]s, 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 (Condition)|Restrained]], [[Prone (Condition)|Prone]], [[Sleeping (Condition)|Sleeping]], [[Entangled (Condition)|Entangled]], [[Paralysed (Condition)|Paralysed]], [[Off Balance (Condition)|Off balance]], [[Enwebbed (Condition)|Enwebbed]], [[Blinded (Condition)|Blinded]].
*Attacking an enemy while being [[Hide|Hidden]] or [[Invisible (Condition)|invisible.]]
*Armour, Weapons, and Spells that grant advantage when attacking enemies of a specific [[Races|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|ranged attack]] against an enemy that is within 5ft and making you [[Threatened (Condition)|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.
{| class="wikitable mw-collapsible <!--mw-collapsed-->"
|+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. On average, the bonus/penalty is +/-3.325. 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{{math|n}} represents an {{math|n}}-sided die, {{math|P(i)}} is the probability that a variable has value {{math|a}}, {{math|\mathbb{E} }} denotes the average or expected value of a roll, and {{math|1=\textstyle\sum_{i=a}^b x_i}} denotes the sum of a series of numbers {{math|x}} over an index {{math|i}} with {{math|i}} going from {{math|a}} through {{math|b}}.

The formula to calculate the expected value, {{math|\mathbb{E}[x]}}, of a variable {{math|x}} is equal to the sum of every possible value of {{math|x}} multiplied by the chance for {{math|x}} to have that value.
In the case of an {{math|n}}-sided die, D{{math|n}}, this becomes:
{{math_block|1=\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 {{math|1=P(i) = 1/n}} for any {{math|i}}, and using {{math|1=\sum_{i=1}^n i = \frac{1}{2}n(n+1) }}, we get 
{{math_block|1=\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 {{math|i}} is equal to the chance that the first die rolls {{math|i}} multiplied by the chance that the second die rolls {{math|i}} or less, multiplied by 2 (because the 2 dice are interchangeable), minus the chance of both dice rolling {{math|i}} (because we counted that possibility twice by multiplying by 2). This gives
{{math_block|1=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{{math|n}} we get
{{math_block|1=\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 {{math|1=\sum_{i=1}^n i^2 = \frac{1}{6}n(n + 1)(2n + 1)}}, which gives
{{math_block|1= \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
{{math_block|1= \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 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 {{math|\{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.

=== Effects on critical successes and failures ===
When making an ability check, attack roll or saving throw, a 1 or a 20 will {{em|always}} be treated as a critical failure or success, respectively, regardless of the results after any potential modifiers are added. On a dice roll without advantage or disadvantage, this effectively means there is a {{math|1/20}} (or 5%) chance of either a critical success or failure.

Having advantage or disadvantage can drastically increase or reduce the chance of critical successes and Failures. For example, when rolling with advantage, the only way to get a Critical Failure is to roll  {{em|two}} 1s at the same time. The odds of this result is {{math|1=1/20 \cdot 1/20 = 1/400}} (or 0.25%). Conversely, rolling a Critical Success is far more likely - out of the 400 possible dice roll outcomes,  39 will result in a 20 (rolling 20 on the first die and 1, 2, 3, ... 20 on the second die, plus rolling 20 on the second die and 1, 2, 3, ... 20 on the first die, minus one so that the result of two 20s is not doubly-counted). The odds of this result is {{math|39/400}} (or 9.75%). The opposite is true for rolling with Disadvantage: a Critical Success has a 0.25% chance and a Critical Failure has a 9.75% chance.

Effectively, rolling with advantage means that critical failures are 20 times {{em|less}} likely and critical successes are almost {{em|twice}} as likely, and the inverse is true for disadvantage.

{| class="wikitable"
|+Chance of Critical Successes and Failures with Advantage and Disadvantage
|-
!Outcome!!Normal Roll!!Roll With Advantage!!Roll With Disadvantage
|-
|Critical Failure (1)||5%||0.25%||9.75%
|-
|Critical Success (20)||5%||9.75%||0.25%
|}

== Similar effects ==
While advantage and disadvantage only apply to {{D20}} rolls, some character features can grant a similar bonus to other rolls. The clearest example is [[Savage Attacker]], a [[feat]] which has a character roll all melee damage dice twice, taking the highest result. This is effectively advantage on melee [[damage rolls]], though none of the advantage rules apply: the effect can stack with others that double damage dice, and there is no similar negative effect that gives you the equivalent of disadvantage on damage.

Using the result of the calculations above to see what the average bonus to our damage becomes, depending on what dice the weapon uses.

{| class="wikitable"
|+ 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 Attack|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 [[Dip#Condition:_Dipped_in_Fire|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==

*[https://www.youtube.com/watch?v=X_DdGRjtwAo The unexpected logic behind rolling multiple dice and picking the highest] by Matt Parker
*[http://onlinedungeonmaster.com/2012/05/24/advantage-and-disadvantage-in-dd-next-the-math/ Advantage and Disadvantage in D&D Next: The Math] by The Online Dungeon Master (Michael Iachini)
*[https://statmodeling.stat.columbia.edu/2014/07/12/dnd-5e-advantage-disadvantage-probability/ D&D 5e: Probabilities for Advantage and Disadvantage] by Bob Carpenter

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