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 or not 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 die:

• d4
• d6
• d8
• d10
• d12
•  d20

The number of dice rolled is notated by with a number placed immediately before the d and any applicable modifiers are given in addition (if it is a bonus) or substraction (if is a penalty) after the dice notation – when a single twenty sided die (d20) is rolled with no modifiers, it is notated as 1d20, when two six-sided dice (d6) are rolled with a modifier of +2, the roll is notated as 2d6+2.

A number of modifiers are potentially added to dice rolls. These modifiers can be either bonuses {{subst:}} which are added to the result – or penalties, which are subtracted from the results. These are notated

The range of results is usually given in parentheses. For example, a single dart from a  Magic Missile spell does 1d4+1 (2-5) Force damage, meaning it rolls 1d4 and adds 1 to the result, giving a possible total of between 2 and 5 points of damage.

Dice rolls are expressed in the game using both dice notation: $xdN+y$, where x is the number of dice rolled, and N is the number of sides of those dice. y is the sum of all modifiers added to the dice roll.

The game also expresses damage using a range: $a\sim b$, where a is the minimum damage possible and b is the maximum damage possible.

Modifiers

A number of modifiers are potentially added to dice rolls. These modifiers can be either bonuses {{subst:}} which are added to the result – or penalties, which are subtracted from the results.

Ability score modifiers

Most rolls have an associated ability and creatures add their corresponding ability score modifier to the outcome of rolls they make.

Creatures also add an ability score modifier to the save DCs of actions they take or spells that they cast, again corresponding to an associated ability.

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.

Creatures also add their proficiency bonus to the save DCs of spells they cast and class actions they take, and to the save DCs of actions they take using weapons they are proficiency with.

Additional modifiers

Some features and conditions add additional modifiers to save DCs and the result 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.

d20 rolls

Whenever a creature attempts an action that has a chance of failure, they roll 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 categorised 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 they make an attack roll against the target's DC 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 attacks.

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 the final result after modifiers are applied. Saving throws attempted during dialogue and death saving throws can also roll 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, however when saving throws are used during dialogue they will guarantee failure or success when a 1 or 20 is rolled.

Advantage and disadvantage

A number of situations and conditions give creatures advantage or disadvantage on d20 rolls. A creatur that makes a roll with advantage rolls two d20 dice separately, and use the higher of the two results. If they instead 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. Likesise, 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, 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.

Neither proficiency nor expertise stack: there's no benefit to having multiple sources of proficiency for a skill, nor is there any benefit to having both proficiency and expertise at the same time. 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.

There is no special benefit to having both proficiency and expertise at the same time – the proficiency bonus is still only doubled. Some sources of expertise do, however, require the character to already be proficient in a skill.

Common scenarios

Automatic rolls

Some ability checks are automatic. For example, when a creature approaches an unactivated 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 representa creature’s attempt to “save” themselves from harm. Spells and actions taken by other creature 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, adding their corresonding ability score modifier to the results.

When attemptint a save, a creature adds an ability score modifier corresponding to that save's associated ability. If they are also proficient in saves made using that ability, they add their proficiency bonus as well.

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

While the outcome 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. Often, the damage or conditions inflicted by the associated effect will be reduced in severity, or sometimes ignored entirely.

A number of features affect saving throws,

Save proficiency

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

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

For the Difficulty class of a spells that can be saved against is determined through the following formula:

8 + proficiency bonus + spellcasting ability modifier.

Weapon save dc

Each weapon action can grant its own inherent bonus DC that isn't listed anywhere, but most frequently it's +2. The Difficulty Class of saves allowed by weapon actions is calculated as follows:

Weapon Action DC = 8 + proficiency bonus + Strength or Dexterity + inherent weapon action bonus DC

Certain weapon actions, notably Concussive Smash, instead use allow the acting creatures to either use their spallcaster 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 spell caster, 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 could have a DC of just 5, whereas an effective trap could have a DC of 15. A slightly spoiled tart could impose a Constitution save of DC 5, 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 and companions after they have been  Downed. Three successful saves will let a creature stabilise, no longer needing to make saves, and three failures will lead to the creature becoming  Dead.

Death saving throws are not associated with an ability score and so don't add any modifiers or the proficiency bonus. They only benefit from bonuses that apply to all saving throws (such as  Bless) or specifically to death saves. Death saves are a success on a result of 10 or higher, and otherwise a failure. A character dies when three failures are accumulated, or becomes Stable (they no longer make death saves) when three successes are accumulated, whichever happens first.

Attack rolls

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

Attack rolls are rolled against the target's AC. If the attempt is successful, the attack hit, and the attacker rolls for damage. If the result was less than the target AC, the attack was a miss.

Attack roll modifiers

Attack rolls are always made using an associated ability:

• Attacks made with melee weapons, including thrown weapons, generally add the attacking creature's Strength modifier, unless they have the Finesse property, in which case they add either the attacker's Strength or Dexterity modifier, whichever is higher.
• 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, they also add their proficiency bonus.

Critical hits

Example of a critical hit with a 1d6 shortsword.

When a creature rolls a natural 20 on an an attack roll, the attack is not only an automatic hit – it is also a critical hit, and the attacker rolls twice the normal number of dice to determine damage dealt, including additional dice such as those from smites or combat manoeuvres. 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 19 or higher on attack rolls. Multiple sources of this effect stack, allowing the critical hit threshold to go even lower than 19.

Some conditions, such as  Sleeping and  Paralysed, automatically turn all incoming attacks into critical hits.

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 have a number 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 default formula that determines AC is:

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

The AC bonus from Dexterity modifier may be capped when wearing medium and is ignored entirely when wearing heavy armour.

Medium armour caps the Dexterity Modifier to +2, whrereas heavy armour ignores the modifier entirely.

Shields will grant the listed AC bonus to whomever equips it, regardless of which weapon slot they are currently using.

Other formulas

Unarmoured creatures may use a different formula if they have certain features. Creatures always use whichever formula – which they have access to – would result in a higher AC.

 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 also given following dice notation: 1d4Piercing damage.

Damage modifiers

Modifiers added to damage rolls are only added once per source, even 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's usually add theability score modifier that they added to the attack's 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 by  Agonising Blast – which allows a warlock to add their Charisma modifier to the damage rolls of their  Eldritch Blast.

Proficiency bonuses are not added to damage rolls.

Other rolls

Healing

Healing restores a target's hit points but otherwise works 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 very low or very high 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 (50% chance to hit) to attack rolls, and deals 2 damage per attack, the defender would on average survive for 10 turns.

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

Meanwhile, if the defender has 19 AC (30% chance to be hit), they would survive for an average of 16.66 rounds. 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, with 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, 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 6.125% 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 is only 4.5, because 2d4 can never roll a 1. 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%	42.25%
10	55%	79.75%	30.25%
11	50%	75%	25%
12	45%	69.75%	20.25%
13	40%	64%	16%
14	35%	57.75%	12.25%
15	30%	51%	9%
16	25%	43.75%	6.25%
17	20%	36%	4%
18	15%	27.75%	2.25%
19	10%	19%	1%
20	5%	9.75%	0.25%

Effects of advantange 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 we get that having advantage is equivalent to an average bonus of +3.325.

Because of symmetry, having disadvantage instead of advantage means we can simply make the permutation of $\{1,\dots ,n\}\to \{n,\dots ,1\}$ 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 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 is $1/20\cdot 1/20=1/400$ (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 $39/400$ (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, and 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