Editing Dice rolls

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| title = Dice rolls
| description = Dice rolls are a central mechanic in Baldur's Gate 3 which determine the outcome of many different situations.
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[[File:Deception.png|thumb|A Deception check]]
'''Dice rolls''' are a central game mechanic in ''Baldur's Gate 3''. Dice are rolled to determine the outcome of variety of situations, such as whether or not a character will succeed at using a particular skill, or if an attack will land and how much damage it will do.
{{TOC|limit=3}}
== Dice notation ==
Dice are notated with a ''d'' followed by the number of sides on that specific dice: {{hlist|
* {{DieIcon|d4|Force}} d4
* {{DieIcon|d6|Radiant}} d6
* {{DieIcon|d8|Cold}} d8
* {{DieIcon|d10|Poison}} d10
* {{DieIcon|d12|Psychic}} d12
* {{D20}}}}

The number of dice to be rolled is notated immediately before the ''d''. Any applicable '''[[#Modifers|modifiers]]''' for the roll are given as an addition (if it is a '''bonus''') or subtraction (if is a '''penalty''') after the dice notation. When a single twenty sided dice (d20) is rolled with no modifiers, it is notated as {{InfoBlob|1d20}}. When two six-sided dice (d6) are rolled with a modifier of +3, the roll is notated as {{InfoBlob|2d6+3}}.

The range of potential results is often given in parentheses, especially for [[#Damage rolls|damage rolls]]. For example, a single dart from a {{SAI|Magic Missile|h=20px}} spell does {{InfoBlob|1d4+1 (2-5) Force damage}}. This means rolling 1d4 and adding 1 to the result, giving a possible total of 2 to 5 points of damage.

== Modifiers ==
A number of '''modifiers''' are potentially added to dice rolls. Modifiers are either '''bonuses''' which add to the result, or '''penalties''' which are subtracted from it. A roll may have bonuses and/or penalties from multiple sources; in such cases they are added together and expressed as a single modifier. For example, a d20 roll with a bonus of +5 and a penalty of -2 would be expressed as {{InfoBlob|1d20+3}}.

; Ability score modifiers : Most rolls have an associated [[ability]], and creatures add their corresponding ability score modifier to the outcome of rolls they make.
; Proficiency bonus : Creatures add their proficiency bonus to any attack rolls, ability checks, or saving throws that they make using weapons, skills, or saves that they are proficient in, as well as to all attack rolls made during spell attacks.
; Additional modifiers : Some [[features]] and [[conditions]] add additional modifiers to save DCs and the results of rolls, such as [[Shillelagh]], which allows the caster to add their spellcasting ability modifier to their attack and damage rolls, instead of Strength or Dexterity.

When a creature forces an opponent to make a [[#Saving throw|saving throw]] against a spell the creature has cast or action they have taken, the applicable modifiers are added to the creature's save DC instead.

== d20 rolls ==
Whenever a creature attempts an action that has a chance of failure, it rolls a twenty-sided die (d20) against a target number to determine whether the attempt was a success or a failure, and add any applicable modifiers. If the result is equal to or exceeds the target number, the attempt was successful. If the result was lower than the target number, or if the creature rolled a 1, the attempt failed.

These attempts are categorized either as attack rolls – which are rolled against the target's [[Armour Class]] (AC), as ability checks – which are rolled against the check's Difficulty Class (DC), or as saving throws – which are rolled against a save DC:

<center>Formula = {{InfoBlob|{{D20}} + Ability score modifier + Proficiency bonus (if proficient) + Other modifiers (if any)}}</center>

; Attack rolls : When a creature attacks a target, it makes an attack roll against the target's AC to determine whether the attack is a hit or a miss. If the attack is a hit, it generally deals damage, and the attacker rolls for damage. Creatures generally make their attacks with their equipped [[weapon]] (including unarmed), but some [[spells]] – such as a [[Warlock]]'s [[Eldritch Blast]] – require the caster to make spell attack rolls.
; Saving throws : Traps, spells, conditions, and other sources of harm may allow a creature a chance to avoid or reduce their effect, known as a saving throw or ''save''. To attempt a save, a creature rolls a d20 against a target save DC.
; Ability checks : An ability check is an attempt to succeed at a specific task, and is rolled against a Difficulty Class (DC) set by the game for that task. If the final result of the roll equals or exceeds the DC, the attempt is successful.

A Difficulty Class (or DC) is a number rolled against when making ability checks or saving throws. It represents how difficult a task is to accomplish.

The number is determined by the the task attempted – or in the case of saves – the spell, condition, or action that has to be overcome.

=== Natural 1s and 20s ===
Rolling a 1 or 20 on a d20 roll is referred to as a ''natural 1'' or ''natural 20''. When making an attack roll or ability check, rolling a natural 1 is always an automatic failure, while a natural 20 is always an automatic success, regardless of the final result after modifiers are applied. Saving throws attempted during dialogue, and death saving throws, can also roll natural 1s and 20s.

Unlike attack rolls and ability checks, saving throws are not guaranteed to fail or succeed when the d20 result is either a natural 1 or 20 respectively, unless they occur during dialog.

=== Advantage and disadvantage ===
[[File:Advantage Icon.png|alt=The in-game symbol for advantage.|left]]
[[File:Disadvantage Icon.png|alt=The in-game symbol for disadvantage|left]]
A [[List of sources of advantage|number of situations and conditions]] give creatures advantage or disadvantage on d20 rolls. A creature that makes a roll with advantage rolls two d20 dice separately, and uses the higher of the two results. If they have disadvantage, they choose the lower of the two.

Creatures receive no additional benefit or penalty from having multiple sources of advantage or disadvantage on a dice roll, and still only roll twice. Likewise, creatures that have {{em|both}} advantage and disadvantage on a roll only roll a single die, even if they have multiple sources of either.

== Ability checks ==
Ability checks are dice rolls made to determine whether a creature succeeds or fails at a task. They are rolled against the task's Difficulty Class (DC), which is generally predetermined by the game. Each ability check is made using one of the six [[abilities]] in the game, and creatures add an ability's corresponding ability score modifier to the results of ability checks they make.

=== Skills ===
Ability checks are usually made using a specified skill. Skills are specific areas of expertise, each associated with an ability, that characters can be proficient in.

Characters add their proficiency bonus to any ability checks they make using skills they are proficient in.{{note|These rolls are often referred to as "skill checks" by the community, although they are not referred to as such in-game.}}

{{SkillsTable}}

All characters gain proficiency in two skills based on their chosen [[background]] during character creation, and can choose 2-4 more skills to be proficient in from a list of skills determined by their [[class]].

Additionally, some [[races]], subclasses, and [[feats]] also give proficiency in specific skills, and [[bards]] receive the class feature [[Jack of All Trades]] at level 2, allowing them to add {{em|half}} their proficiency bonus (rounded down) to ability checks they make using skills they {{em|are not}} proficient in.

Proficiency does not stack – there's no benefit to having multiple sources of proficiency for a skill.

==== Expertise ====
[[File:Expertise.png|right]]
Characters can also have expertise in a skill, which allows them to add {{em|double}} their proficiency bonus when making a corresponding ability check. While it is possible to have proficiency {{em|and}} expertise in a skill at the same time, they do not stack. Some sources of expertise do, however, require the character to already be proficient in a skill.

Sources of expertise that require prior proficiency in the respective skill include:
* [[Rogue]]s gain expertise in any two skills they are proficient in at both level 1 and level 6.
* [[Bard]]s gain expertise in any two skills they are proficient in at both level 3 and level 10.

Sources of expertise that {{em|do not}} require prior proficiency in the respective skill include:
* [[Knowledge Domain]] [[Cleric]]s can select two skills to master amongst [[Arcana]], [[History]], [[Nature]] and [[Religion]] at level 1.
* The [[Actor]] feat gives expertise in [[Deception]] and [[Performance]].
* [[Gnome#Rock Gnomes|Rock Gnomes]] have expertise in [[History]].
* The [[Illithid Expertise]] feature grants expertise in [[Persuasion]], [[Deception]], and [[Intimidation]]. 

=== Common scenarios ===
; Automatic rolls
: Some ability checks are automatic. For example, when a creature approaches an inactive trap, the game rolls a [[Perception]] ability check to determine whether the creature notices the trap. Perception is a Wisdom skill, so the character adds their Wisdom modifier and, if proficient in Perception, their proficiency bonus to the ability check. Once the trap is discovered, the character can interact with it to attempt to [[Disarm]] it, which requires a successful [[Sleight of Hand]] check, a Dexterity skill.
; During dialogue
: Ability checks are also common during dialogue, where some responses require an ability check to determine the outcome. Examples include using Charisma-based skills like [[Persuasion]], [[Deception]], or [[Intimidation]] to influence others, or Intelligence-based skills like [[Investigation]], [[History]], or [[Religion]] to determine or remember facts.
; Contests
: A contest is a special type of ability check in which two creatures both roll an ability check to oppose each other, and one wins over the other. The creatures don't necessarily roll the same type of check.
:: An example of this is the {{SAI|Shove}} action. The creature attempting the Shove rolls Athletics, and the defending creature rolls either Athletics or Acrobatics (the game chooses the Skill with the highest bonus) to contest the Shove. If the attacker's roll is higher than the defender's, the Shove succeeds; otherwise, it fails.

== Saving throws ==
Saving throws represent a creature’s attempt to “'''save'''” themselves from harm. Spells and actions taken by other creatures frequently allow their targets to attempt a save, as do hazards like [[traps]] and [[surface]]s. Each save has an associated ability – referred to using terms like '''Strength saving throw''' or '''Dexterity save''' – and a save DC that creatures attempting to save roll against. When attempting a save, a creature adds an ability score modifier corresponding to that save's associated ability, and if they are proficient in saves made using that ability, they add their proficiency bonus as well.

While the result of an attempted saving throw is always binary – it is either a success or a failure – the exact outcome of a successful save depends on the effect in question. Typically, the damage or conditions inflicted by the associated effect will be reduced in severity, and sometimes negated entirely.

Saving throws do not automatically fail or succeed on natural 1s and 20s, except when made during dialogue.

A [[List of features and items that affect saving throws|number of features]] affect saving throws, and some races have advantage on certain saves.

=== Save proficiency ===
All classes give save proficiency with two abilities. Though when multiclassing, only the '''first''' class taken gives its save proficiencies. An additional save proficiency can be gained by taking the [[Resilient]] feat.

{{ClassSavingThrowsTable}}

=== Save DCs ===
The Difficulty Class rolled against when attempting to save is called ''save DC''. A successful save can mean completely avoiding negative effects, reducing the damage received (usually by half), or both. For example, successfully saving against a spike trap could mean that a creature takes no damage at all, because it successfully evaded the spikes. On the other hand, if it's caught in the area of effect of a {{SAI|Fireball}}, then a successful save will merely halve the damage. Saving against {{SAI|Thunderwave}} both halves the damage taken, and prevents a creature from being pushed by the spell.

Different mechanics calculate save DC differently:
; Danger save DC : In scenarios such as traps, the game chooses an appropriate Difficulty Class, depending on how serious the danger is. This includes consumable items such as elemental arrows or throwables.
; Spell save DC : The Difficulty Class of a spell that can be saved against is determined through the following formula:
:: {{InfoBlob|8 + proficiency bonus + spellcasting ability modifier}}.
: Certain [[conditions]] and [[List of Equipment that Affect Spell DC|equipment]] worn by the caster can also affect their Spell Save DC.
; Weapon save DC: Most weapons allow proficient users to perform special "weapon actions", which are typically limited to once per short rest (e.g. [[Backbreaker]]). These actions often include the chance to inflict a condition on the target, and these conditions require the target to attempt a Save to avoid them. Each weapon action can grant its own inherent bonus to DC that isn't listed anywhere, but is frequently +2. The Difficulty Class of saves allowed by weapon actions is calculated as follows: 
:: {{InfoBlob|1=Weapon Action DC = 8 + proficiency bonus + [[Strength]] or [[Dexterity]] modifier + inherent weapon action bonus DC}}
: Certain weapon actions, notably [[Concussive Smash]], instead allow the acting creature to either use their Spell Save DC or weapon action DC with a +2 bonus, whichever is higher.

==== Other effects ====
In the case of threats that don't originate from a spellcaster, such as a trap or a poisonous apple, the game sets the DC based on how serious the threat is intended to be. For example, a rather ineffective trap might have a DC of just 5, whereas an effective trap could have a DC of 15. A slightly spoiled tart could impose a DC 5 Constitution save when eaten, whereas a potent venom from a snake could impose a DC 15 Constitution save on the victim.

=== Death saving throws ===
Death saving throws are a special type of saving throw made by playable characters after they have been {{cond|Downed}}. Death saves are made once per turn while the character remains Downed. If a Downed character receives damage from any source that isn't a critical hit, they automatically fail one death saving throw. A critical hit against a Downed character results in 2 failed saves. Melee attacks against a Downed target are always classified as a critical hit. 

Three successful saves will let a creature stabilize, no longer needing to make death saves to survive, and three failures will lead to the creature becoming {{cond|Dead}}. 

Death saving throws are not associated with an ability score and so don't get any modifiers, nor do they benefit from the proficiency bonus. They only benefit from bonuses that apply to all saving throws (such as {{SAI|Bless}}) or specifically to death saves (such as [[Family Ring]]). Death saves are always DC 10. A character dies when three failures are accumulated, or stabilizes when three successes are accumulated, whichever happens first.

Death saving throws can be critical failures and critical successes. A natural 1 rolled for a death save will add ''two'' failures to a character's death save count, while a natural 20 will immediately stabilize the character regardless of their current death save count.

== Attack rolls ==
Creatures make attack rolls when they attack a target, usually with a [[Weapon|weapon]] or a [[Spells|spell]].

If the result of the attack roll is equal to or higher than the target's Armour Class (AC), the attack ''hits'', and the attacker rolls for damage. If the result is lower than the target's AC, the attack ''misses''.

=== Attack roll modifiers ===
Attack rolls are always made using an associated ability:
* Unarmed attacks, and attacks made with melee weapons and thrown weapons generally add the attacking creature's Strength modifier. 
* If the weapon has the [[Finesse]] property, attacks with it add either the attacker's Strength or Dexterity modifier, whichever is higher.
* Unarmed attacks may use the attacker's Dexterity modifier if they have certain features like [[Martial Arts: Dextrous Attacks|Dextrous Attacks]] from the [[Monk]] class.
* Attacks made with ranged weapons add the creature's Dexterity modifier.
* Spell attacks add the caster's [[Spells#Spellcasting|spellcasting ability modifier]], generally determined by their [[Class|class]].

If the attacker is proficient with the weapon they are wielding, or if the attack is a spell attack or unarmed attack, they also add their proficiency bonus.

=== Critical hits ===
{{main|Critical hit}}
[[File:critical hit.png|right|thumb|Example of a critical hit with a 1d6 shortsword.]]
When a creature rolls a natural 20 on an attack roll, the attack is a [[critical hit]]''.'' Critical hits automatically land regardless of the target's AC, and the attacker also rolls twice the normal number of dice to determine damage dealt, including additional dice such as those from smites or combat maneuvers. Modifiers and bonuses – including the creature's relevant ability score modifier and proficiency bonus – are not doubled.

Some [[feats]], [[Classes|class features]], and [[items]] [[Critical Hit threshold reduction|reduce]] the critical hit threshold by 1, allowing the creature to land critical hits by rolling either 19 or 20 on attack rolls. Multiple sources of this effect stack, allowing the critical hit threshold to go even lower than 19.

=== Armour Class ===
Armour Class (AC) is a measurement of how difficult a creature is to be hit by an attack. In order to successfully hit a creature, the results of an [[Attack roll|attack roll]] must be equal to or greater than the target's Armour Class. AC can be increased by equipping [[Armour|armour]] and [[Shields|shields]], by selecting certain [[Feats|feats]] when leveling up, or utilizing certain [[Spells|spells]].

==== Formula ====
The formula that determines AC when wearing Armour in the torso slot is:
: {{InfoBlob|Torso armor AC + [[Dexterity|Dexterity modifier]] + shield bonus + other bonuses and penalties}}
The AC bonus from Dexterity is typically capped at +2 when wearing [[Armour#Medium armour|medium armour]]{{note|The [[Medium Armour Master]] feat increases the cap from +2 to +3.}}{{note|A few rare armours have an "Exotic Material" trait that allow the wearer to get the full Dexterity bonus to AC. These include [[Yuan-Ti Scale Mail]], [[Unwanted Masterwork Scalemail]], [[Sharpened Snare Cuirass]], and [[Armour of Agility]].}}, and is reduced to zero when wearing [[Armour#Heavy armour|heavy armour]].

Most [[Shields]] grant +2 AC.

Other bonuses include things like the [[Fighting style|Defense]] fighting style, which grants +1 AC while wearing armor, and the [[Cloak of Protection]], which grants +1 AC at all times. Bonuses to AC stack with each other.

==== Other formulas ====
[[Barkskin]] sets the affected creature's AC to 16 if they would otherwise have less.

Unarmoured creatures may use a different formula if they have certain features. Creatures always use whichever formula they have access to that would result in the highest AC. Alternative formulas are only used if no items marked "Light Armor", "Medium Armor", or "Heavy Armor" are being worn in any equipment slot.

{{SAI|Mage Armour}} and {{SAI|Draconic Resilience}}:
: {{InfoBlob|13 + Dexterity modifier + shield bonus + other bonuses and penalties}}

{{SAI|Unarmoured Defence (Barbarian)}}:
: {{InfoBlob|10 + Constitution modifier + Dexterity modifier + shield bonus + other bonuses and penalties}}

{{SAI|Unarmoured Defence (Monk)}}:
: {{InfoBlob|10 + Wisdom modifier + Dexterity modifier + other bonuses and penalties}}

== Damage rolls ==
The base damage dealt by a [[weapons|weapon]], [[spells|spell]], class action, or condition is usually determined by a ''damage roll''. Damage rolls always have an associated [[damage type]] that is given following the dice notation, e.g. {{DamageText|1d4|Piercing}}.

=== Damage modifiers ===
Modifiers added to damage rolls are only added {{em|once}} per source, even if multiple dice are rolled.

Which ability score modifier is added to a damage roll depends on the attack:
* When making weapon attacks, the attacking creature usually adds the ability score modifier that they added to the attack roll.
* Ability score modifiers are not normally added to damage rolls dealt by spells or spell attacks, unless specifically stated otherwise in the spell's description, or if enabled by some feature, such as {{SAI|Agonising Blast|h=20px}}.

Proficiency bonuses are {{em|not}} added to damage rolls unless the attack being used (e.g. [[Shadowsoaked Blow]]) says so.

== Other rolls ==
; Healing : [[Healing]] restores a target's [[hit points]] in a similar manner to damage rolls. Healing rolls may also add modifiers, but there's no general rule for this; any bonuses are determined by the source of the healing. For example, a [[Potion of Healing]] restores {{DamageText|2d4+2|Healing}}. There are many magic items, class features, and other effects which also provide bonuses to healing, for example the {{Class|Life Domain}}'s {{SAI|Disciple of Life}} feature.
; Wild Magic : When a Wild Magic sorcerer casts a leveled spell, a d20 is rolled to determine if they will trigger a Wild Magic Surge. A surge is triggered only when the outcome is 20. The resulting effect, and Wild Magic Barbarian surge effects for Rage: Wild Magic, are also determined with dice rolls.

== Karmic Dice ==
[[File:karmic dice setting.png|upright=0.7|thumb|The optional Karmic Dice setting, located in Gameplay Options]]
When the Karmic Dice option is enabled (it is by default), the game will avoid streaks of low rolls. 

However, Karmic Dice influences all rolls – including those of enemies – and the results always skew toward a positive result for the dice roller. In short, '''the Karmic Dice setting makes combat encounters quicker and deadlier''' for both you and your enemies, as attacks are more likely to hit and do higher damage. 

Karmic Dice was previously referred to as "Loaded Dice".

== Mathematics ==
A wide variety of mathematics can be applied to understand dice roll mechanics in greater depth.

=== Armour Class mathematics ===
Armour Class becomes more useful the greater it is – the difference in effectiveness between 20 and 19 AC is {{em|greater}} than the difference in effectiveness between 15 and 14.

To illustrate this, if a defender has 15 AC and 10 HP, and the attacker has +5 to attack rolls, and deals 2 damage per attack, the defender would on average survive for 10 turns because the attack has a 50% chance to hit against 15 AC.

If the defender's AC was increased to 16 (chance to hit drops to 45%), they would instead survive for an average of 11.1 rounds (an 11% increase in effectiveness).

Meanwhile, if the defender starts with 19 AC (30% chance to be hit), they would survive for an average of 16.66 rounds. But if their AC was increased to 20 (25% chance to be hit), they would be able to survive for an average of 20 rounds (a 20% increase in effectiveness).

The difference between 25 and 24 is even greater, granting a {{em|200%}} increase in effectiveness (50 vs 100 rounds).

=== Damage rolls mathematics ===
Note that due to the mathematics of dice rolls, the difference between, say, 1d8 and 2d4 is more than just the higher minimum value of 2 on the 2d4 roll. With the d8, you have an equal chance of getting, say, a 5 and an 8.  On the other hand, the 2d4 roll is statistically more likely to lead to a total value of 5, rather than a total value of 8. This is most easily explained with a table of all possible outcomes:

{| class="wikitable" style="text-align: center;"
|+ Possible results of a 2d4 roll, highlighting the number of possibilities resulting in a total value of 5
|-
! First roll !! Second roll !! Total value
|-
| 1 || 1 || 2
|-    
| 1 || 2 || 3
|-    
| 1 || 3 || 4
|-    
| 1 || 4 || {{color|red|'''5'''}}
|-    
| 2 || 1 || 3
|-    
| 2 || 2 || 4
|-    
| 2 || 3 || {{color|red|'''5'''}}
|-    
| 2 || 4 || 6
|-    
| 3 || 1 || 4
|-    
| 3 || 2 || {{color|red|'''5'''}}
|-    
| 3 || 3 || 6
|-    
| 3 || 4 || 7
|-    
| 4 || 1 || {{color|red|'''5'''}}
|-    
| 4 || 2 || 6
|-    
| 4 || 3 || 7
|-    
| 4 || 4 || 8
|}

Notice how often the 5 appears in the possibilities for the '''total value''' (4 out of 16 possibilities) vs. how often the 8 appears (1 out of 16).  This means a 2d4 roll has a 25% chance of resulting in 5 points of damage, but only a 6.25% chance of resulting in 8 points of damage.  Meanwhile, the 1d8 roll actually has a higher chance of resulting in the maximum damage value of 8, since 1 out of 8 possibilities (12.5%) result in an 8. However, the average roll of 2d4 is 5 damage, while the average roll of 1d8 is only 4.5. Therefore, 2d4 is generally more consistent in damage output and will result in higher rolls in the long run. 

=== Advantage mathematics ===
==== Effects of advantage on success ====
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%|| 36%
|-
|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 of advantage 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{{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, the results is 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 of advantage on criticals ====
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, while 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%
|}

==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

== Footnotes ==
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{{NavGameplay/Mechanics}}