Advantage: Difference between revisions

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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.

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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.

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==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 "

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| 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 in formulas: ''P(a)'' is the chance that a variable has value ''a'', ''_adv'' means "with advantage", and ''Σ(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''. For example, the sum of all numbers on a D6 would be written as ''Σ(i=1,6)(i)''.

The formula to calculate the average, ''<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. ''<x> = Σ(i=a,b)(x_i * P(x_i))''.

To calculate the average of any die, Dx, this becomes:

* ''<Dx> = Σ(i=1,x)(i * P(i))''

For a regular dice roll the chance distribution is homogenous, which means ''P(i) = 1/x'' for any ''i'', and using ''Σ(i=1,n)(i) = n * (n+1)/2'', we get

* ''<Dx> = Σ(i=1,x)(i * P(i)) = x * (x + 1)/2 * 1/x = (x + 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_adv(i) = 2 * P(i) * Σ(j=1,i)(P(j)) - P(i)² = 2 * 1/x * i/x - 1/x² = (2i - 1)/x²''

Applying that to the formula of an average of a die Dx we get

* ''<Dx>_adv = Σ(i=1,x)(i * (2i - 1)/x²) = 2/x² * Σ(i=1,x)(i²) - 1/x² * Σ(i=1,x)(i)''

Here we can use that the sum of squares is ''Σ(i=1,n)(i²) = n * (n + 1) * (2n + 1)/6'', which gives

* ''<Dx>_adv = 2/x² * x * (x + 1) * (2x + 1)/6 - 1/x² * x * (x + 1)/2 = 2x/3 + 1 + 1/3x - 1/2 - 1/2x = 2x/3 + 1/2 - 1/6x''

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

* ''<Dx>_adv - <Dx> = 2x/3 + 1/2 - 1/6x - (x + 1)/2 = (x - 1/x)/6''

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 ... x} → {x ... 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 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 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==

Advantage: Difference between revisions

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