Dice rolls

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 a character will succeed at using a particular skill, or if an attack will land and how much damage it will do.

Dice notation

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

• d4
• d6
• d8
• d10
• d12
•  d20

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

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 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 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:

Formula =  d20 + Ability score modifier + Proficiency bonus (if proficient) + Other modifiers (if any)

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 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

A 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 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.

Strength Athletics

Dexterity Acrobatics Sleight of Hand Stealth

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
Strength	Athletics
Dexterity	Acrobatics Sleight of Hand Stealth
Intelligence	Arcana History Investigation Nature Religion
Wisdom	Animal Handling Insight Medicine Perception Survival
Charisma	Deception Intimidation Performance Persuasion

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 half their proficiency bonus (rounded down) to ability checks they make using skills they are not proficient in.

Proficiency does not stack – there's 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 proficiency 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:

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

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

• Knowledge Domain Clerics can select two skills to master amongst Arcana, History, Nature and Religion at level 1.
• The Actor feat gives expertise in Deception and Performance.
• 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  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 surfaces. 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 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.

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 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  Fireball, then a successful save will merely halve the damage. Saving against  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:

8 + proficiency bonus + spellcasting ability modifier.

Certain conditions and 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:

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  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  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  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 or a 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 Dextrous Attacks from the Monk class.
• Attacks made with ranged weapons add the creature's Dexterity modifier.
• Spell attacks add the caster's spellcasting ability modifier, generally determined by their 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 article: Critical hit

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, class features, and items 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 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 leveling up, or utilizing certain spells.

Formula

The formula that determines AC when wearing Armour in the torso slot is:

Torso armor AC + Dexterity modifier + shield bonus + other bonuses and penalties

The AC bonus from Dexterity is typically capped at +2 when wearing medium armour, and is reduced to zero when wearing heavy armour.

Most Shields grant +2 AC.

Other bonuses include things like the 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.

 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

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 following the dice notation, e.g. 1d4Piercing.

Damage modifiers

Modifiers added to damage rolls are only added 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  Agonising Blast.

Proficiency bonuses are 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 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 many magic items, class features, and other effects which also provide bonuses to healing, for example the  Life Domain's  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

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 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 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:

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 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.

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 $\textstyle \sum _{i=a}^{b}x_{i}$ 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

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

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 will 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 $1/20$ (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 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 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