Dice rolls

A Deception check

Dice rolls are a central game mechanic in Baldur's Gate 3. When the player character attempts most actions, a roll helps resolve the outcome based on the relevant DC or AC and any modifiers for the acting character. Player choices decide the approach, while a roll determines the end result, such as whether an ability check succeeds, an attack hits, or how much damage is dealt.

Dice notation

Dice are notated with a d followed by the number of sides on that specific die:

• d4
• d6
• d8
• d10
• d12
•  d20

The number of dice to be rolled is notated immediately before the d. Any applicable modifiers to the roll are given as an addition (for a bonus) or subtraction (for a penalty) after the dice notation. For example, when a single twenty-sided die is rolled with no modifiers, it is notated as 1d20. When two six-sided dice are rolled with a modifier of +3, the roll is notated as 2d6+3.

The range of potential results is often given in parentheses, especially for damage rolls. For example, a single dart from a  Magic Missile spell does 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. Note that when rolling multiple dice, not all numbers in the range are equally likely outcomes, as explained under damage roll mathematics.

In some instances (such as attack rolls), the game displays a percentage denoting the chance of success, calculated from modifiers and other factors as explained in this article. A check with a Difficulty Class of 15 and a roll of 1d20+4 (ranging from 5 to 24) shows a 50% chance of success (rolls from 11 to 20 yielding a minimum of 15).

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 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 is expressed as 1d20+3.

Ability score modifiers

Most rolls have an associated ability, and creatures add their corresponding ability score modifier to the outcome of their rolls. Some features change what modifier is used for a particular type of roll. For example, the spell  Shillelagh has the caster add their spellcasting ability modifier to their attack and damage rolls with the weapon it is cast upon, instead of Strength or Dexterity.

Proficiency bonus

Creatures add their proficiency bonus to any attack rolls, ability checks, or saving throws that they make using weapons, skills, or saves they are proficient in. Proficiency bonus is also added to all spell attack rolls.

Additional modifiers

Some features and conditions add additional modifiers to the results of rolls. For example,  Guidance adds 1d4 to the result of ability checks made by the recipient.

d20 rolls

Whenever a creature attempts an action that has a chance of failure, it rolls a d20 against a target number to determine whether the attempt will succeed or fail, and then adds any applicable modifiers. If the result is equal to or exceeds the target number, the attempt is successful. If the result is lower than the target number (or for some rolls, if the unmodified dice roll is 1), the attempt fails.

These attempts are categorised as attack rolls – rolled against the target's Armour Class (AC), ability checks – rolled against the check's Difficulty Class (DC), or saving throws – rolled against a save DC. In general, most d20 rolls follow a similar formula:

 d20 + ability score modifier + proficiency bonus (if proficient) + other modifiers

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. The weapon they are using (or lack thereof) determines their ability score modifier, normally Strength or Dexterity. If the attack is a hit, the attacker rolls for damage. Creatures usually make attacks with their equipped weapon, but some spells – such as  Eldritch Blast – require the caster to make spell attack rolls; a spell attack functions similarly to a weapon attack, but uses the caster's spellcasting ability to determine the ability score modifier.

Saving throws

Traps, spells, conditions, and other sources of harm may allow a creature the chance to avoid or reduce their effects, known as a saving throw or save. To attempt a save, a creature rolls a d20 against a target save DC. The ability associated with the saving throw is determined by the type of harm caused by the source. For a physical threat that must be avoided, a Dexterity saving throw is made. For a poisonous effect, a Constitution saving throw is made. For a magical or non-corporeal effect, the saving throw may be Intelligence, Wisdom, or Charisma, though most commonly Wisdom.

Ability checks

An ability check is an attempt to perform a specific task, and the associated ability and DC are 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 and saving throws. It represents how difficult a task is to accomplish.

The number is determined by the task being attempted – or in the case of saving throws – the spell, condition, or action which must be overcome.

Natural 1s and 20s

Rolling an unmodified 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.

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

Advantage and disadvantage

Many situations and conditions give creatures advantage or disadvantage on d20 rolls. A creature that makes a roll with advantage rolls two d20s separately, and uses the higher of the two results. If they have disadvantage, they use 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. In addition, if creatures have both advantage and disadvantage on a roll, these conditions cancel each other out, and only a single d20 is rolled, even if they have multiple sources of either.

For an exhaustive list of sources of advantage and disadvantage, see the following pages:

• For attack rolls: Advantage and Disadvantage
• For saving throws: Advantage and Disadvantage

When determining a passive skill value, having advantage or disadvantage on ability checks with the given skill adds or subtracts 5 from the value. Passive skill values are used during contests.

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 DC. The DC of these checks are 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.

Strength Athletics

Dexterity Acrobatics Sleight of Hand Stealth

Constitution None

Intelligence Arcana History Investigation Nature Religion

Wisdom Animal Handling Insight Medicine Perception Survival

Charisma Deception Intimidation Performance Persuasion

List of skills, sorted by ability

Ability Score	Skills
Charisma	Deception Intimidation Performance Persuasion
Constitution	None
Dexterity	Acrobatics Sleight of Hand Stealth
Intelligence	Arcana History Investigation Nature Religion
Strength	Athletics
Wisdom	Animal Handling Insight Medicine Perception Survival

Custom characters gain proficiency in two skills based on their chosen background during character creation, with origin characters having predetermined backgrounds. All characters can choose 2 more skills to be proficient in from a list of skills determined by their class (bards and rangers choose 3, while rogues choose 4).

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

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

Expertise

Characters can also have expertise in a skill, which allows them to add double their proficiency bonus when making a corresponding ability check. While it is possible to have both proficiency and expertise in a skill, they do not stack. However, some sources of expertise do require the character to already be proficient in a skill.

Sources of expertise that require prior proficiency in the respective skill include:

• Bards gain expertise in any two skills they are proficient in at both level 3 and level 10.
• Rogues gain expertise in any two skills they are proficient in at both level 1 and level 6.

Sources of expertise that do not require prior proficiency in the respective skill include:

• A Rock gnome's expertise in History
• A Knowledge Domain Cleric's selection of expertise in two of the following skills at level 1: Arcana, History, Nature, or Religion
• Expertise in Deception and Performance from the  Actor feat
• Expertise in Deception, Intimidation, and Persuasion from the  Illithid Expertise feature

Types of ability checks

Automatic rolls

Some ability checks are automatic. For example, when a creature approaches a trap, a passive Perception check is rolled to determine whether they notice it. Perception is a Wisdom skill, so the creature adds their Wisdom modifier and, if proficient in Perception, their proficiency bonus to the ability check.

During dialogue

Ability checks are also common during dialogue, where some responses require an ability check to determine the outcome. Examples include using Charisma skills like Persuasion, Deception, or Intimidation to influence others, or Intelligence skills like Arcana, History, or Religion to determine or remember facts. Before these checks, a button labelled "Add Bonus" appears along the bottom of the screen, allowing the player character and other party members to freely use available spells and consumables to grant bonuses to the check. This is also possible when making ability checks while interacting with objects, such as when disarming traps or picking locks.

Contests

A contest is a special type of ability check where one creature rolls an ability check against another creature's skill. Rather than rolling, the opposing creature sets the DC for the ability check using their passive skill value, given by 10 + skill modifier. If a creature currently has a source of advantage on ability checks with the relevant skill, they add 5 to their passive skill; with disadvantage, they subtract 5.

Contests are used when attempting to  Shove or  Throw another creature. The creature attempting the shove or throw makes a contested Athletics check versus the target's passive Athletics or Acrobatics, whichever is higher.

Contests are also used when  Hiding. The sneaking creature makes a contested Stealth check against an opponent's passive Perception while in their cone of vision to avoid being detected.

Saving throws

Saving throws represent a creature's attempt to "save" themself from harm. Spells and actions taken by other creatures frequently allow their targets to attempt a save, as do hazards like traps and surfaces. Each save has an associated ability and a save DC that creatures attempting to save roll against. When attempting a save, a creature adds the ability score modifier of 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 – 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, or sometimes negated entirely.

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

A number of features affect saving throws, and some races have advantage on certain types of saves.

Save proficiency

Each class grants proficiency in saving throws with two abilities. Though when multiclassing, only the first class taken grants saving throw proficiencies (unless they come from a separate feature, like  Iron Mind). An additional save proficiency can be gained by taking the Resilient feat.

Saving Throw Proficiencies by Class

Barbarian	Bard	Cleric
Strength  Constitution	Dexterity  Charisma	Wisdom  Charisma
Druid	Fighter	Monk
Intelligence  Wisdom	Strength  Constitution	Strength  Dexterity
Paladin	Ranger	Rogue
Wisdom  Charisma	Strength  Dexterity	Dexterity  Intelligence
Sorcerer	Warlock	Wizard
Constitution  Charisma	Wisdom  Charisma	Intelligence  Wisdom

Save DCs

The Difficulty Class rolled against when attempting to save is called the 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 because it successfully evades the spikes. On the other hand, if caught in the area of effect of a  Fireball spell, then a successful save merely halves the damage. Saving against a  Thunderwave spell both halves the damage taken and prevents a creature from being pushed by the spell.

Different mechanics calculate save DC differently:

Spell save DC

The Difficulty Class of a spell that can be saved against is determined through the following formula:

8 + proficiency bonus + spellcasting ability modifier

Each class has a spellcasting ability, which is used for its class features (features that appear under a class's tab in the spellbook) and spells that call for a spellcasting ability modifier. Some non-class features also use this spellcasting ability, such as spells cast through items (including scrolls), free recasts of Recastable spells (regardless of source), and Illithid Powers. A multiclassed character uses the spellcasting ability of their newest class (the class in which they most recently reached level 1) for non-class features. There are some class features which are bugged and will incorrectly use the newest class's spellcasting ability, as noted on those features' pages.

Spellcasting Ability

Intelligence	Wisdom	Charisma
Fighter  Rogue  Wizard	Cleric  Druid  Monk  Ranger	Barbarian  Bard  Paladin  Sorcerer  Warlock

Certain conditions and equipment worn by the caster can also affect their spell save DC.

Spellcasting classes can check their spell save DC in the spellbook (shortcut "K"). A target's saving throws can be checked by examining them (shortcut "T").

Weapon action 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 is not listed anywhere, but is frequently +2. The Difficulty Class of saves against weapon actions is calculated as follows:

Weapon Action DC = 8 + proficiency bonus + Strength or Dexterity modifier (whichever is higher) + inherent weapon action bonus DC

Equipment other than weapons may have features that use weapon action DC. Some class features also use weapon action DC, notably most manoeuvres of a  Battle Master, and many  Monk abilities, such as  Stunning Strike.

Some unique weapons actions instead use spell save DC.

Hybrid save 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.

Fixed Save DC

In the case of threats that do not originate from a spellcaster, such as a trap or poisonous apple, the game sets the DC based on how serious the threat is intended to be. For example, a slightly spoiled tart could impose a DC 5 Constitution save when eaten, whereas a potent snake venom could impose a DC 15 Constitution save on the victim. Many consumable items, such as elemental arrows, throwables, and coatings, use fixed save DCs.

Death saving throws

Death saving throws are a special type of saving throw made by playable characters after they have been  Downed. Death saves are made once per turn while the character remains Downed. Death saves are always DC 10. If a Downed character receives damage from any source that is not 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 critical hits.

Playable characters  die when three failures are accumulated or stabilise when three successes are accumulated, whichever happens first.

Death saving throws are not associated with an ability score and so do not get any ability score modifiers, nor do they benefit from a proficiency bonus. They only benefit from bonuses which apply to all saving throws (such as  Bless) or specifically to death saves (such as from the  Family Ring).

Attack rolls

Creatures make attack rolls when they attack a target, usually with a weapon or a spell.

If the result of the attack roll is equal to or higher than the target's AC, the attack hits, and the attacker rolls for damage, unless prevented by immunity or another feature. If the result is lower than the target's AC, the attack misses unless it's a critical hit.

Attack roll modifiers

Attack rolls are always made using an associated ability:

• Unarmed attacks, melee weapon attacks, and thrown weapon attacks generally add the attacking creature's Strength modifier.
• Attacks made with ranged weapons add the creature's Dexterity modifier.
• Attacks from weapons with the  Finesse property or monk weapons and unarmed attacks from creatures with  Martial Arts: Dextrous Attacks, add either the attacker's Strength or Dexterity modifier, whichever is higher.
• Spell attacks add the caster's spellcasting ability modifier, generally determined by their class.

Spell attacks, unarmed attacks, and attacks with weapons that the attacker is proficient with also add their proficiency bonus. Certain attacks add extra modifiers as well:

• If the attacker is using an enchanted weapon, the enchantment value is added, from +1 to +3.
• According to high ground rules, attacking from at least 2.5 m / 8 ft above applies a +2 bonus to the attack roll, while attacking from at least 2.5 m / 8 ft below applies a -2 penalty.

Attack rolls are also modified by:

• Class Features - e.g.  Archery
• Coatings - e.g.  Diluted Oil of Sharpness
• Conditions - e.g.  Lightning Charges
• Feats - e.g.  Sharpshooter
• Items - e.g.  Gloves of Dexterity
• Spells - e.g.  Bless

For a comprehensive list, see list of features and items that affect attack rolls.

Critical hits

Example of a critical hit with a 1d6 shortsword.

Main article: Critical hit

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 manoeuvres. However, some items (e.g.  Adamantine Shield) grant immunity to critical hits, treating them as regular attack rolls.

Modifiers and flat bonuses (i.e. any bonus that doesn't involve dice) – including the creature's relevant ability score modifier – are not doubled.

Some feats, class features, and items reduce the critical hit threshold by 1, allowing attack rolls to land critical hits by rolling either 19 or 20. Multiple sources of this effect stack, allowing the critical hit threshold to go below 19.

Critical misses

When a creature rolls a natural 1 on an attack roll, the attack is a critical miss. Critical misses automatically fail regardless of the target's AC or attack roll modifiers.

Armour Class

Armour Class (AC) is a measurement of how difficult a creature is to hit with an attack. In order to successfully hit a creature, the results of an attack roll must be equal to or greater than the target's Armour Class. AC can be increased by equipping armour and shields, by selecting certain feats when levelling up, or by utilising certain spells.

Base formula

The formula which determines AC when not wearing armour is:

10 + Dexterity modifier + shield bonus + other bonuses and penalties

If armour is equipped, then its AC value will be used instead of 10. The AC bonus from Dexterity is typically capped at +2 when wearing medium armour, and is removed entirely when wearing heavy armour.

Most shields grant +2 AC, though some rare ones grant +3 AC.

Other bonuses include things like the  Defence fighting style, which grants +1 AC while wearing armour, and the  Cloak of Protection, which grants +1 AC at all times. AC bonuses stack with each other.

Other formulas

Unlike bonuses to AC, which all stack with each other, some spells and features grant a creature a new base formula with which to determine AC when not wearing armour. Only one of these formulas can be active at a time, and if a creature has access to multiple, then it will use whichever results in the highest AC at the given time. Any piece of equipment tagged Light Armour, Medium Armour, or Heavy Armour will impede the use of any of these formulas when equipped, even if it provides no AC value itself.

 Mage Armour and  Draconic Resilience:

13 + Dexterity modifier + shield bonus + other bonuses and penalties

 Unarmoured Defence (Barbarian):

10 + Constitution modifier + Dexterity modifier + shield bonus + other bonuses and penalties

 Unarmoured Defence (Monk):

10 + Wisdom modifier + Dexterity modifier + other bonuses and penalties

The spell  Barkskin sets the affected creature's AC to 16 if they would otherwise have less. It is not impeded by wearing armour.

Damage rolls

The base damage dealt by a weapon, spell, class action, or condition is usually determined by a damage roll. Damage rolls always have an associated damage type that is given in dice notation, e.g. 1d4⁠⁠Piercing or 2d6⁠⁠Fire.

Damage modifiers

Modifiers added to damage rolls are only added once per source, even if multiple dice are rolled.

Whether an ability score modifier is added to a damage roll depends on the attack:

• When making weapon attacks, the attacking creature usually adds the same ability score modifier to both the attack roll and damage roll.
• Ability score modifiers are not normally added to damage rolls dealt by spells or spell attacks, unless specifically enabled by a feature such as  Agonising Blast.

Proficiency bonuses are not added to damage rolls unless the attack being used (e.g.  Shadowsoaked Blow) states otherwise.

Other rolls

Healing

Healing restores a target's hit points similarly 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 2d4+2⁠⁠hit points. There are magic items, class features, and other effects which provide bonuses to healing, for example, the  Life Domain Cleric's  Disciple of Life feature.

Wild Magic

When a  Wild Magic Sorcerer casts a levelled 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 is also determined with a dice roll, as are the similar  Wild Magic Barbarian surge effects for  Rage: Wild Magic.

Karmic Dice

The optional Karmic Dice setting, located in Gameplay Options

When the Karmic Dice option is enabled (as it is by default), the game mitigates bad-luck streaks (e.g., many missed attacks in a row). It applies to all combatants, including NPCs and enemies, as well as ability checks made while in dialogue. It does not apply to ability checks made outside of dialogue, such as Sleight of Hand checks when picking locks or Perception checks to discover hidden objects. It does not penalise lucky streaks. Larian described using Karmic Dice as akin to having a "friendly DM". Some players report that this can make fights feel quicker or more potent, but results can vary.

Internally, the game keeps track of the probability, $p$, to make any given roll, and if the roll is failed, this amount is added to a "debt". This debt is tracked separately for all characters, including party members, allies, and enemies. If this debt is positive before any given roll, there is a chance equal to $\frac{p}{1-\text{Debt}}$ to automatically pass it. If passed, an amount equal to $2(1-p)$ is subtracted from the debt total. Otherwise, the roll is resolved as usual. Note that for this purpose, succeeding a saving throw against an enemy spell counts as a failed roll for the enemy (and vice versa). Therefore, Karmic Dice may lead to more failed saving throws than usual.

A secondary effect from enabling Karmic Dice is that damage rolls are made against a ⁠Gaussian distribution, with a mean equal to $\frac{D+1}{2}$ and a standard deviation equal to $(D-1)(\frac{1}{3}-\frac{D}{120})$ for a $D$-sided damage die, rather than a normal uniform distribution. The effect of this is to reduce the variance of the rolls and skew them to the mean. Thus, certain features, such as  Great Weapon Fighting and  Savage Attacker, may be less effective when Karmic Dice are enabled.

Mathematics

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

Armour Class mathematics

Armour Class becomes increasingly useful (in an increasingly non-linear fashion) the greater it is. For example, the increase in effectiveness from 19 to 20 AC is greater than the increase in effectiveness from 16 to 17. To illustrate this:

• If a defender has 16 AC and 10 HP, and the attacker has +5 to attack rolls, and deals 2 damage per attack, the defender will survive for 10 turns on average, because the attack has a 50% chance to hit. If the defender's AC was increased to 17 (chance to hit drops to 45%), they will instead survive for an average of ~11.11 rounds - an improvement of ~11.1%.
• If the defender starts with 19 AC (35% chance to be hit), they will survive for an average of ~14.29 rounds. But if their AC is increased to 20 (30% chance to be hit), they will be able to survive for an average of ~16.67 rounds (a ~16.67% increase in effectiveness).
• The difference from 24 to 25 is even greater, granting a 100% increase in effectiveness (50 vs 100 rounds).

Damage roll mathematics

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, one has 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:

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	5
2	1	3
2	2	4
2	3	5
2	4	6
3	1	4
3	2	5
3	3	6
3	4	7
4	1	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 results 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 one needs 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.

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$n$ represents an $n$-sided die, $P(i)$ is the probability that a variable has value $a$, $\mathbb{𝔼}$ denotes the average or expected value of a roll, and ${\sum }_{i=a}^b{x}_i$ denotes the sum of a series of numbers $x$ over an index $i$ with $i$ going from $a$ through $b$.

The formula to calculate the expected value, $\mathbb{𝔼}[x]$, 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.

In the case of an $n$-sided die, D$n$, this becomes:

$\mathbb{𝔼}[\text{D}n]=\sum _{i=1}^n(i\cdot P(i))$

For a regular dice roll the probability distribution is uniform, which means $P(i)=1/n$ for any $i$, and using $\sum _{i=1}^{n}i=\frac{1}{2}n(n+1)$, we get

$\mathbb{𝔼}[\text{D}n]=\sum _{i=1}^n(i\cdot P(i))=\frac{1}{n}\left(\frac{n(n+1)}{2}\right)=\frac{n+1}{2}$

For a dice roll with advantage the chance to roll the number $i$ 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

$P_{\text{adv}}(i)=2P(i)\sum _{j=1}^{i}P(j)-P(i{)}^2=2\frac{1}{n}\cdot \frac{i}{n}-\frac{1}{n^2}=\frac{2i-1}{n^2}$

Applying that to the formula of an average of a die D$n$ we get

$\mathbb{𝔼}[\text{D}n\text{ with advantage}]=\sum _{i=1}^{n}i\cdot \frac{2i-1}{n^2}=\frac{2}{n^2}\cdot \sum _{i=1}^n{i}^2-\frac{1}{n^2}\cdot \sum _{i=1}^{n}i$

Here we can use that the sum of squares is $\sum _{i=1}^n{i}^2=\frac{1}{6}n(n+1)(2n+1)$, which gives

$\mathbb{𝔼}[\text{D}n\text{ with advantage}]=\frac{2}{n^2}\left(\frac{n(n+1)(2n+1)}{6}\right)-\frac{1}{n^2}\left(\frac{n(n+1)}{2}\right)=\frac{2n}{3}+1+\frac{1}{3n}-\frac{1}{2}-\frac{1}{2n}=\frac{2n}{3}+\frac{1}{2}-\frac{1}{6n}$

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

$\mathbb{𝔼}[\text{D}n\text{ with advantage}]-\mathbb{𝔼}[\text{D}n]=\frac{2n}{3}+\frac{1}{2}-\frac{1}{6n}-\frac{n+1}{2}=\frac{1}{6}(n-\frac{1}{n})$

When we apply this expression to a d20, the result 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 $\{1,\dots ,n\}\to \{n,\dots ,1\}$ 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 critical rolls

When making an ability check, attack roll, or saving throw, a 1 or a 20 is (almost) always treated as a critical failure or success, respectively, regardless of the results after any potential modifiers are added. On die rolls without advantage or disadvantage, this effectively means there is a $1/20$ (or 5%) chance of either a critical success or failure.

Having advantage or disadvantage has significant impact on increasing or decreasing the chance of critical successes and Failures. For example, when rolling with advantage, the only way to get a Critical Failure is to roll two 1s at the same time. The odds of this result are $1/20\cdot 1/20=1/400$ (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 are $39/400$ (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 less likely, and critical successes are almost twice as likely, while the inverse is true for disadvantage.

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

• The unexpected logic behind rolling multiple dice and picking the highest (Video) by Matt Parker
• Advantage and Disadvantage in D&D Next: The Math (Blog with Text and Tables) by The Online Dungeon Master (Michael Iachini)
• D&D 5e: Probabilities for Advantage and Disadvantage (Blog with Graphs and Tables) by Bob Carpenter
• How good is Advantage in D&D? (Video) by Joseph Newton