Dice expression calculator

What the calculator works out

Players and game masters keep asking the same two questions. What does this roll average out to, and what are the odds of reaching a number? And what does this character really do in a round: how often does the attack hit, how often is it a critical, and how much damage is that on average? This page answers both. The first tab reads any dice expression and counts every possible outcome. The second combines the d20 attack roll against Armor Class with your damage dice, including advantage, disadvantage, an improved critical range and several attacks per round.

The method

For a plain roll of n dice with s sides, the calculator adds one die at a time to the list of probabilities of each total. For the best or worst k of n dice it counts the ways the dice can fall by face value, with the multinomial weights, so that 4d6 drop lowest comes out at 12.2446 on average (15,869/1,296), matching the exact tables of the dice probability calculator. For an attack, with a d20 roll and a bonus, a roll of 1 misses, a roll of 20 (or your critical range) is a critical hit, and any other roll hits if the roll plus the bonus is at least the target's Armor Class (SRD 5.1). Advantage takes the better of two d20 rolls, so the chance of the face v is (2v − 1)/400; disadvantage uses (41 − 2v)/400. Damage is the mixture of “miss: 0”, “hit: normal damage” and “critical: damage dice doubled”.

Worked example: a longsword at +5 against AC 15

A d20 roll of 10 or more hits (10 + 5 = 15), and a 20 is a critical, so the chance to hit is 55%, of which 5% is critical. A normal hit with 1d8+3 averages 7.5; a critical, 2d8+3, averages 12.0. Average damage per attack: 50% × 7.5 + 5% × 12.0 = 4.35, with a standard deviation of 2.29 on a hit alone. With advantage the hit chance is 79.75% and the average damage 6.42; with disadvantage the hit chance falls to 30.25%. A Champion's critical range of 19–20 doubles the critical chance to 10% and raises the average to 4.575 without changing the chance to hit. A rogue who adds Sneak Attack dice (1d8+3+3d6) averages 10.65 per attack.

Common mistakes and limits

Doubling the modifier on a critical. Only the dice are doubled. In the SRD the modifiers are added once, so a +3 stays +3: 2d8+3, not 2d8+6.

Stacking advantage. Two sources of advantage still mean one extra d20, and advantage plus disadvantage cancel out completely, however many sources there are of each (SRD 5.1). Choose only one of the three in the calculator.

Forgetting that 1 and 20 override the numbers. A natural 1 misses even against AC 1 and a natural 20 hits even against AC 40. The calculator applies this: with a +50 bonus against AC 10, the chance to hit is still 95%, not 100%.

Reading the average as what happens. The average is a long-run figure over many rounds. A single round with a few attacks is far from it: the standard deviation of one attack is larger than its average here, so check the table of damage and the chance to deal at least the target's hit points.

Limits. Saving throws, resistance, rerolls and special class features are not modeled. The notation with kh, kl, dh and dl is common among online dice rollers, not part of the SRD.

What dice notation does the calculator understand?

Terms joined by + or −. A term is a plain number (3), a roll such as 2d6, d20 (one die) or d% (a d100), or a roll with a keep or drop suffix: kh3 keeps the 3 highest dice, kl2 the 2 lowest, dl drops the lowest (dl2 drops two), dh drops the highest. So 2d6+3 is two six-sided dice plus 3, 4d6dl1 is the ability-score method of rolling four dice and dropping the lowest, and 2d20kh1 is a d20 roll with advantage. Up to 100 dice in all, with up to 1,000 sides and a largest possible total of 20,000; keep and drop work for up to 20 dice with up to 100 sides each.

How does advantage change the odds? What is the average d20 with advantage?

With advantage you roll two d20 and use the higher one; with disadvantage, the lower one (SRD 5.1). The average d20 is 10.5, with advantage 13.825, with disadvantage 7.175. Against AC 15 with a +5 bonus, one d20 hits 55% of the time, advantage 79.75%, disadvantage 30.25%. If a roll has both advantage and disadvantage, you have neither and roll one d20, no matter how many of each there are; two sources of advantage still give only one extra die.

What are the chances of a critical hit?

A natural 20 on one d20 is 5%. With advantage the chance of at least one 20 in two rolls is 1 − 0.95² = 9.75%, with disadvantage 0.05² = 0.25%. A fighter with the Champion archetype scores a critical on 19 or 20 (Improved Critical), which is 10% on one die, or 19.00% with advantage; a higher-level champion on 18–20 (Superior Critical), 15%. A natural 20 always hits and a natural 1 always misses whatever the bonus or the armor class, so even a +20 attacker misses 5% of the time.

How does a critical hit change the damage?

In the SRD you roll all of the attack's damage dice twice and add them together, then add the modifiers once. A longsword attack of 1d8+3 becomes 2d8+3 on a critical: 7.5 on average becomes 12.0. Extra dice, such as a rogue's Sneak Attack, are doubled too, so 1d8+3+3d6 averages 18.0 on a normal hit and 33.0 on a critical. Damage with a keep or drop term is rolled twice as a whole. Some monsters have their own rule, for example one in the SRD that rolls three times.

How much damage will an attack do on average?

Average damage = chance of a normal hit × average normal damage + chance of a critical × average critical damage. For +5 against AC 15 with a 1d8+3 weapon, that is 50% × 7.5 + 5% × 12.0 = 4.35 per attack. With advantage it rises to 6.42, and with a Champion's 19–20 critical range to 4.575. Multiply by the number of attacks for the damage per round. This is a long-run average; the calculator also shows the standard deviation, because a single round varies a lot.

How is this different from the dice probability calculator?

The dice probability calculator takes n identical dice and gives the odds of a sum, the sum of the best k and the chance of rolling a face. This one reads a full expression with several terms and modifiers, gives the whole distribution of the result, and has an attack tab that combines the d20 roll against Armor Class with the damage roll. If you only need the odds of a sum of identical dice, the other tool is quicker.

What does the calculator not model?

Anything that is not an attack roll followed by a damage roll: saving throws and spell effects with half damage on a save, resistance and vulnerability, rerolls (such as the Great Weapon Fighting style), exploding dice and extra attacks that depend on hitting. Negative damage totals are counted as 0 here. The probability of every result is exact for the dice you enter, but whether a rule applies at your table is for you and your game master.

Sources: System Reference Document 5.1, Wizards of the Coast, the sections “Advantage and Disadvantage”, “Rolling 1 or 20”, “Critical Hits” and the Champion fighter archetype (Improved Critical, Superior Critical); Wikipedia, “Dice notation” for the NdS notation and the 4d6-drop-lowest odds. This work includes material taken from the System Reference Document 5.1 (“SRD 5.1”) by Wizards of the Coast LLC and available at https://dnd.wizards.com/resources/systems-reference-document. The SRD 5.1 is licensed under the Creative Commons Attribution 4.0 International License available at https://creativecommons.org/licenses/by/4.0/legalcode. Related: dice probability calculator, expected value calculator, probability calculator and streak probability calculator. How we check our calculators.