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Advantage: Difference between revisions
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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. | 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: | 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= | 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 | 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 Dx we get | Applying that to the formula of an average of a die Dx 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= | 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}{n} = \frac{2n}{3} + \frac{1}{2} - \frac{1}{6n} }} | |||
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{x + 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'''. | When we apply this elegant expression to a D20 we get that '''having advantage is equivalent to an average bonus of +3.325'''. | ||
Revision as of 21:15, 23 November 2023
Advantage and Disadvantage are a gameplay mechanic that can greatly affect the success of dice rolls. They can apply to Attack Rolls, Saving Throws, and Ability Checks, but not to Damage Rolls.
Advantage
When you roll with Advantage, you perform the roll twice, and use the higher result. It doesn't stack beyond 2 dice, regardless of how many sources of advantages you have.
Example: You roll two d20 for an Attack Roll, the results are 16 and 4. Your effective result is 16.
On a D20, having advantage raises the average of your roll by 3.325 to 13.825 (for the math, see below)
Advantage and Disadvantage cancels each other, and having multiple sources doesn't change this either. Even if you have three sources of Advantage, a single source of Disadvantage will cancel it.
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 perform the roll twice, and use the lower result. It doesn't stack beyond 2 dice, regardless of how many sources of disadvantages you have.
Example: You roll two d20 for an Attack Roll, because you have Disadvantage. 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)
Disadvantage and Advantage cancel each other, and having multiple sources doesn't change this either. Even if you have three sources of Disadvantage, a single source of Advantage will cancel it.
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.
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.
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 represents an -sided die, is the probability that a variable has value , denotes the average or expected value of a roll, the subscript "adv" means "with advantage", and denotes the sum of a series of numbers over an index with going from through .
The formula to calculate the expected value, , of a variable is equal to the sum of every possible value of multiplied by the chance for to have that value. In the case of an -sided die, D, this becomes:
For a regular dice roll the probability distribution is uniform, which means for any , and using , we get
For a dice roll with advantage the chance to roll the number is equal to the chance that the first die rolls multiplied by the chance that the second die rolls or less, multiplied by 2 (because the 2 dice are interchangeable), minus the chance of both dice rolling (because we counted that possibility twice by multiplying by 2). This gives
Applying that to the formula of an average of a die Dx we get
Here we can use that the sum of squares is , which gives
To know what bonus having advantage gives to our roll, we calculate
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 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 means you have advantage on your 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.
- Weapon deals 1d4 damage: average bonus damage is 0.625
- Weapon deals 1d6 damage: average bonus damage is 0.972...
- Weapon deals 1d8 damage: average bonus damage is 1.3125
- Weapon deals 1d10 damage: average bonus damage is 1.65
- Weapon deals 1d12 damage: average bonus damage is 1.9861...
- Weapon deals 2d6 damage: average bonus damage is 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