Advantage: Difference between revisions
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{{NoExcerpt|''For a comprehensive summary of the mechanics behind all rolls and modifiers, see: [[Die Rolls]]''}} | {{NoExcerpt|''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 | {{Advantage}} and {{Disadvantage}} are a gameplay mechanic 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 == | == 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 | 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. | Example: You roll two {{D20}} for an Attack Roll, the results are 16 and 4. Your effective result is 16. | ||
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'''Having advantage raises the average of your roll by 3.325 to 13.825.''' (For the math, see below.) | '''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 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.]] | *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]]. | *Armour, Weapons, and Spells that grant advantage when attacking enemies of a specific [[Races|Race]]. | ||
==Disadvantage== | ==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.) | |||
'''On a D20, having disadvantage lowers the average of your roll by 3.325 to 7.175''' ( | |||
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]]. | * 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. | *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== | ==Math== | ||
===Chances of succeeding a specific roll=== | === 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. | 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-->" | {| class="wikitable mw-collapsible <!--mw-collapsed-->" | ||
|+ Chance of rolling a target number or above on 1d20 | |+Chance of rolling a target number or above on 1d20 | ||
|- | |- | ||
! Target on 1d20 !! Normal Roll !! Roll With Advantage !! Roll With Disadvantage | !Target on 1d20!!Normal Roll!! Roll With Advantage!!Roll With Disadvantage | ||
|- | |- | ||
| 1 || 100% || 100% || 100% | |1|| 100% || 100%||100% | ||
|- | |- | ||
| 2 || 95% || 99.75% || 90.25% | |2||95%|| 99.75%||90.25% | ||
|- | |- | ||
| 3 || 90% || 99% || 81% | |3||90% ||99%||81% | ||
|- | |- | ||
| 4 || 85% || 97.75% || 72.25% | |4 ||85%|| 97.75% ||72.25% | ||
|- | |- | ||
| 5 || 80% || 96% || 64% | | 5||80%|| 96%||64% | ||
|- | |- | ||
| 6 || 75% || 93.75% || 56.25% | |6||75% || 93.75%|| 56.25% | ||
|- | |- | ||
| 7 | |7 | ||
| 70%||91%||49% | |||
|- | |- | ||
| 8 || 65% || 87.75% || 42.25% | |8||65% ||87.75%||42.25% | ||
|- | |- | ||
| 9 || 60% || 84% || 42.25% | |9||60% ||84%|| 42.25% | ||
|- | |- | ||
| 10 || 55% || 79.75% || 30.25% | |10||55%||79.75%||30.25% | ||
|- | |- | ||
| 11 || 50% || 75% || 25% | |11||50%||75%||25% | ||
|- | |- | ||
| 12 || 45% || 69.75% || 20.25% | |12 ||45%|| 69.75%||20.25% | ||
|- | |- | ||
| 13 || 40% || 64% || 16% | |13||40%||64%||16% | ||
|- | |- | ||
| 14 || 35% || 57.75% || 12.25% | |14||35%|| 57.75% || 12.25% | ||
|- | |- | ||
| 15 || 30% || 51% || 9% | |15||30% ||51%||9% | ||
|- | |- | ||
| 16 || 25% || 43.75% || 6.25% | |16||25% ||43.75%||6.25% | ||
|- | |- | ||
| 17 || 20% || 36% || 4% | |17||20% ||36%||4% | ||
|- | |- | ||
| 18 || 15% || 27.75% || 2.25% | |18||15% ||27.75%||2.25% | ||
|- | |- | ||
| 19 || 10% || 19% || 1% | |19||10% ||19%||1% | ||
|- | |- | ||
| 20 || 5% || 9.75% || 0.25% | |20||5%|| 9.75%||0.25% | ||
|} | |} | ||
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{{math_block|1=\mathbb{E}[\text{D}n] = \sum_{i=1}^n (i \cdot P(i))}} | {{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 | 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} }} | {{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<nowiki>}</nowiki> }} | ||
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 | 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} }} | {{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<nowiki>}</nowiki> }} | ||
Applying that to the formula of an average of a die D{{math|n}} we get | 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}} | {{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 | 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} }} | {{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<nowiki>}</nowiki> }} | ||
To know what bonus having advantage gives to our roll, we calculate | 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) }} | {{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) }} | ||
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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. | 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. | ||
==Application: [[Savage Attacker]]== | ==Application: [[Savage Attacker]] == | ||
The Savage Attacker feat essentially gives advantage to melee damage rolls. | 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. | 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. | ||
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|+ Expected Bonus Damage from Savage Attacker | |+ Expected Bonus Damage from Savage Attacker | ||
|- | |- | ||
! Weapon Die !! Average Bonus Damage | !Weapon Die !!Average Bonus Damage | ||
|- | |- | ||
| 1d4 || 0.625 | |1d4 ||0.625 | ||
|- | |- | ||
| 1d6 || 0.972 | | 1d6|| 0.972 | ||
|- | |- | ||
| 1d8 || 1.3125 | |1d8||1.3125 | ||
|- | |- | ||
| 1d10 || 1.65 | |1d10||1.65 | ||
|- | |- | ||
| 1d12 || 1.9861 | |1d12 ||1.9861 | ||
|- | |- | ||
| 2d6 || 1.94 | |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. | 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. | ||