D&D Dice Probability Calculator: Hit Chances & Damage Stats

Calculate the exact probability of any D&D dice roll — to-hit chances, average damage, advantage benefit, and crit fishing odds.

Calculate Your D&D Dice Odds

Master your D&D combat decisions with statistically backed probability calculations. Below, our interactive calculator computes hit chances, advantage benefits, and average damage for any roll in 5e or d20 systems.

Probability Calculator

Click calculate to see results

Understanding D&D Probability Basics

D&D 5e uses a d20 system. Each roll has 20 equally likely outcomes (5% per number). To calculate hit chance: (21 - Target Number) × 5%. To hit AC 15 with +5 modifier, target number is 10. Hit chance = (21-10) × 5% = 55%. The natural 1 always misses; the natural 20 always hits and crits.

The Power of Advantage

Advantage rolls 2d20 and takes the higher result. The math works as: P(hit with advantage) = 1 - P(miss)². If your normal hit chance is 55%, advantage gives you 1 - 0.45² = 79.75%. Advantage is mathematically equivalent to roughly +5 to your roll. Always seek advantage in combat — Faerie Fire, flanking variants, or class abilities.

Critical Hit Probability

Normal: 5% (rolling natural 20). Advantage: 1 - 0.95² = 9.75%. Elven Accuracy (triple advantage from 3d20): 1 - 0.95³ = 14.26%. Champion Fighter level 3+ (crit on 19-20): 9.75% normal, 18.55% advantage. The difference is huge for crit-fishing builds with paladin smites or rogue sneak attacks.

Damage Probability

Average dice values: d4 = 2.5, d6 = 3.5, d8 = 4.5, d10 = 5.5, d12 = 6.5. A 2d6 attack averages 7 damage. Maximum damage on critical hits doubles dice but not modifier — so a 2d6+3 hit averages 10 damage, while a 2d6+3 crit averages 17 damage (4d6 + 3 = 14+3 = 17, since average 4d6 = 14).

Saving Throw Probabilities

For spellcaster save DCs, calculate the inverse of your hit chance. Spell DC 15 vs Constitution save +5: target rolls 10+ to succeed (55% save chance), so 45% chance of failure. For multi-target spells like Fireball, expected damage is the sum across all targets — failed save = full damage, successful = half damage.

Lucky Feat & Halfling Luck

Halfling Luck rerolls natural 1s. Effective d20 distribution: 1 has same chance as 2, others normal — boosts average roll by 0.475. Lucky feat (3 rerolls per long rest) provides advantage on selected attacks/saves. Mathematically, Lucky's 3 rerolls per day are worth roughly +5 to those specific rolls.

Sharpshooter & Great Weapon Master Math

The famous "-5 to hit, +10 damage" feats benefit you when your hit chance is high enough. Threshold formula: (Damage gain × hit chance after penalty) > (Damage loss × hit chance before). Generally, take the -5/+10 only when you have advantage or hit on a 7+ before the penalty. Use this calculator to model expected damage with and without the penalty for your specific encounter.

Average Damage Per Round (DPR)

DPR is the standard metric for build comparison. Calculate as: (Hit chance × Average damage) + (Crit chance × Crit bonus damage). A Paladin smiting on a hit at 70% hit rate with Divine Smite adding 2d8 deals roughly +9 damage per attack on a 70% hit rate, plus the crit chance doubling the smite dice. Optimization specialists like TreantMonk use DPR to evaluate every build choice.

Roll Multiple Dice Statistics

For multiple dice, use Anydice.com or our calculator above. Common distributions:

  • 4d6 drop lowest: Average = 12.24, used for ability scores
  • 3d6 ability score: Average = 10.5, classic D&D rolling
  • 2d20 take highest (advantage): Average = 13.83
  • 2d20 take lowest (disadvantage): Average = 7.18

Roll dice for free with our Dice Roller and track combat encounters with the Initiative Tracker. For full character building, see our D&D Character Builder Guide.

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Frequently Asked Questions

How is advantage calculated mathematically?

Advantage rolls 2d20 and takes the higher result. P(hit with advantage) = 1 - P(miss)². If your hit chance is 55%, advantage = 1 - 0.45² = 79.75%. Advantage is mathematically equivalent to approximately +5 to your roll for typical AC ranges.

What's the chance of rolling a critical hit?

Natural 20 = 5% normally. With advantage, 9.75% (1 - 0.95²). With Elven Accuracy (triple advantage), 14.26%. Champion Fighter level 3+ crits on 19-20, doubling these numbers (9.75% normal, 18.55% advantage). Half-Orc level 6 fighter with Champion + Elven Accuracy reaches massive crit rates.

Should I take Sharpshooter or Great Weapon Master?

These feats trade -5 hit for +10 damage. Take them only when your hit chance is 70%+ before the penalty (typically with advantage or vs low AC). For typical AC 15 targets, you need a +12 attack bonus to break even. Calculator above shows expected damage with and without the penalty for your specific stats.

How accurate is our probability calculator?

Mathematically exact for normal d20 rolls and advantage/disadvantage. The calculator uses standard probability formulas: P(hit) = (21-target)/20. Advantage = 1 - P(miss)². Damage averages = (faces+1)/2 per die. Critical hits double dice but not modifiers per 5e rules.

What's the average damage of 2d6+4?

Average d6 = 3.5, so 2d6 = 7 average. Plus modifier 4 = 11 average damage per hit. Maximum = 16 (rolling 12 + 4). On a critical hit, dice double but not modifier: 4d6+4 averages 18 damage. Use the calculator above for any dice combination.

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How dice probability works — the math behind D&D 5e rolls

Every roll in Dungeons & Dragons Fifth Edition uses standard dice notation in the form XdY+Z, where X is the number of dice, Y is the number of faces, and Z is an optional flat modifier. The two key statistical properties you need are the expected value and the variance. For any roll of XdY the expected value is E[XdY] = X × (Y + 1) / 2 and the variance is Var[XdY] = X × (Y² − 1) / 12. These formulas are derived from the discrete uniform distribution and are documented in standard probability theory references (see Wikipedia: Dice notation) and dice-specific calculator engines like AnyDice, built by Jasper Flick.

Two concrete worked examples explain why this matters at the table. First, rolling 1d20 for an attack roll gives an expected value of (1 + 20) / 2 = 10.5 and a variance of (400 − 1) / 12 ≈ 33.25, meaning the standard deviation is roughly ±5.77. That huge spread is why a single d20 attack feels swingy and unfair. Second, rolling 3d6 for ability scores gives the same expected value of 10.5 but a variance of only 3 × 35 / 12 = 8.75 (standard deviation ≈ 2.96). The flatter distribution is why 3d6 scores cluster tightly around 10–11 and why the alternative 4d6-drop-lowest method (PHB p.13) was introduced to skew results upward.

The advantage and disadvantage mechanics from the Player's Handbook (PHB p.7, "Advantage and Disadvantage") are technically order-statistic distributions over 2d20. With advantage you roll 2d20 and keep the higher (max), with disadvantage you keep the lower (min). The probability of rolling at least one natural 20 with advantage rises from a baseline 5% (1/20) to 1 − (19/20)² = 9.75%. Conversely, the chance of rolling a natural 1 with disadvantage jumps from 5% to 9.75%, which is why "rolled with disadvantage" feels disproportionately painful.

Most common dice-probability scenarios in D&D 5e play

  1. To-hit roll vs Armor Class. A +5 attack bonus against AC 15 hits on a 10–20, so the base hit chance is 11/20 = 55%. With advantage, the miss chance squares to (9/20)² = 20.25%, giving a 79.75% hit rate. This is the single most important calculation for combat-focused PCs.
  2. Ability score generation (4d6 drop lowest). Per PHB p.13, the expected value rises from 10.5 to roughly 12.24 and the chance of rolling an 18 jumps from 0.46% (3d6) to about 1.62% per stat. Generating six scores gives roughly a 9.3% chance that at least one will be an 18.
  3. Critical-hit probability with Champion Fighter. Standard crit range is a natural 20 (5%). At Champion Fighter level 3, this expands to 19–20 (10%), and at level 15 it expands further to 18–20 (15%), as documented in PHB p.72 under Improved Critical.
  4. Saving throws against the Fireball spell (PHB p.241). Against a target with a +2 Dex save and a spell DC of 15, the save succeeds on a 13–20, which is 8/20 = 40%. The damage is halved on success, so expected damage from 8d6 (average 28) becomes 0.6 × 28 + 0.4 × 14 = 22.4.
  5. Healing potions and standard rest healing. A potion of healing rolls 2d4+2, expected value (2 × 2.5) + 2 = 7. A short-rest Hit Die for a barbarian (d12) plus CON +3 averages 6.5 + 3 = 9.5 HP per die.

Quick-reference table: dice statistics for D&D 5e

Roll Expected Value Variance Std Dev Min–Max Typical Use (PHB ref)
1d42.51.251.121–4Dagger damage (PHB p.149)
1d63.52.921.711–6Shortsword (PHB p.149)
1d84.55.252.291–8Longsword (PHB p.149)
1d105.58.252.871–10Halberd (PHB p.149)
1d126.511.923.451–12Greataxe (PHB p.149)
1d2010.533.255.771–20Attack roll, save, check (PHB p.7)
2d67.05.832.422–12Greatsword (PHB p.149)
3d610.58.752.963–18Classic ability score (PHB p.13)
4d6-drop-low~12.24~5.6~2.373–18Standard ability score (PHB p.13)
8d628.023.334.838–48Fireball damage (PHB p.241)
2d20 advantage~13.83~22.4~4.731–20Advantage rolls (PHB p.7)
2d20 disadvantage~7.17~22.4~4.731–20Disadvantage rolls (PHB p.7)

Formulas: E[XdY] = X(Y+1)/2; Var[XdY] = X(Y²−1)/12. Verifiable on AnyDice using output 3d6 or output 2d20kh1 for advantage.

Pro DM tips for dice probability at the table

  • Set DCs to the 65% range for average-stat parties. A DC of 13 against a +3 modifier hits 55% of the time. The Dungeon Master's Guide (DMG p.238, Typical Difficulty Classes) lists Easy DC 10, Medium DC 15, Hard DC 20, Very Hard DC 25, Nearly Impossible DC 30 — align fail probability to dramatic stakes.
  • Use the Bless spell aggressively. +1d4 to attack rolls (avg +2.5) increases a 55% hit chance to roughly 67%. Pair with advantage and the effective chance climbs into the 85% range.
  • Roll 2d6 instead of 1d12 for "feels-better" damage. Both have similar averages (7 vs 6.5) but 2d6 clusters near 7 and feels less swingy — a tip popularized by community designers and reflected in the Greatsword vs Greataxe debate.
  • Treat advantage and disadvantage as multiplicative on success rates. If your base success probability is p, advantage gives 1 − (1−p)² and disadvantage gives p². Memorize the table: 25% → 43.75%/6.25%, 50% → 75%/25%, 75% → 93.75%/56.25%.
  • Inspiration is worth roughly +1.5 expected value on a key d20 roll. Average d20 result is 10.5, and re-rolling the lower of two d20s yields an expected ~13.83 — about a 3.3-point improvement at the most critical moment.
  • Track failure cascades on three-check skill chains. Three consecutive 65% checks succeed completely only 27.5% of the time. Consider using the group check rule (PHB p.175) or stepping down DCs after partial failures.
  • Print or pin anydice.com outputs for the next big fight. Drop in output [highest 1 of 2d20] + 7 to plan exactly how often your BBEG hits AC 17 with advantage.

Common mistakes when reading dice probabilities

  • Confusing expected value with most likely value. The expected value of 2d6 is 7 and 7 is also the modal result, but for 1d20 the expected value is 10.5 and every individual outcome is equally likely — do not assume "around 10 or 11" without checking the distribution.
  • Assuming dice "balance out" within a short session. Variance scales with the square root of the number of rolls, not linearly. Twenty d20 rolls can absolutely return zero 20s and three 1s. The dice have no memory; this is the gambler's fallacy.
  • Ignoring crit-fail house rules. RAW (rules-as-written, PHB p.194) a natural 1 only auto-misses attacks — it does not auto-fail saves or ability checks. Many home tables forget this distinction.
  • Double-counting advantage with Bardic Inspiration or Bless. The bonuses stack but the advantage mechanic does not stack with itself. Two sources of advantage give the same effect as one source.
  • Mis-applying the "exploding dice" optional rule. If you let max-rolls re-roll (a homebrew popular from systems like Savage Worlds), the expected value of a d6 jumps from 3.5 to 4.2, and damage outputs balloon. Test on AnyDice before introducing it.

Complementary tools and physical dice picks

For deeper probability analysis beyond a simple calculator, the gold standard is AnyDice.com by Jasper Flick — it supports custom dice, conditional rerolls, and per-outcome graphs. For browser-based rolling integrated with a virtual tabletop, Roll20's built-in dice macros and Quantum Roll certified randomness handle most use cases. Dedicated GM tools like Foundry VTT include a robust Dice So Nice module for 3D dice physics. For physical play, weighted-balanced dice from manufacturers documented in independent test publications remove the manufacturing variance that plagues cheap d20s — a balanced d20 should produce each face within ~1% of uniform over 10,000 rolls.

Try AnyDice for advanced probability analysis

Free, web-based, and supports custom dice, advantage, exploding rules, and conditional outputs. Paste output 2d20kh1 + 7 to visualize a +7 attack with advantage instantly.

Open AnyDice

FAQ — dice probability for D&D 5e

What does the dice notation 3d6+2 actually mean?

Roll three six-sided dice, sum the pips, and add 2. The minimum result is 1+1+1+2 = 5 and the maximum is 6+6+6+2 = 20. Expected value = 3 × (6+1)/2 + 2 = 12.5.

How likely is a critical hit on a d20?

5% (1 in 20) for a standard character. A Champion Fighter at level 3 expands the crit range to 19–20 (10%), and at level 15 to 18–20 (15%), per PHB p.72.

Does advantage stack with the Bless spell?

Yes, the +1d4 from Bless (PHB p.219) is a flat additive bonus while advantage is a separate mechanic of rolling 2d20 and keeping the highest. They stack — an attack with both rolls 2d20 keep highest, then adds +1d4 once.

What is the expected value of 4d6 drop the lowest?

Approximately 12.24, compared to 10.5 for 3d6. This is why PHB p.13 recommends this method for player characters — it biases ability scores upward without making 18s common.

How does advantage actually change my hit chance?

If your base hit probability is p, advantage gives 1 − (1 − p)². A 55% base hit becomes 79.75%; a 75% base hit becomes 93.75%. Disadvantage gives p², so 55% becomes 30.25%.

What is the probability of rolling a natural 20 with advantage?

9.75%. Calculated as 1 − (19/20)² = 1 − 0.9025 = 0.0975 or about 1 in 10.25 rolls.

Do critical hits double the damage dice or the total damage?

Per PHB p.196, you double the dice but not modifiers. A longsword crit (1d8 + 4 STR) rolls 2d8 + 4 = 13 average, not 2 × (4.5 + 4) = 17. This distinction matters for high-modifier builds.

What is the chance the Fireball spell kills a 22 HP target with no save bonus?

Fireball is 8d6 = 28 avg, half on save (14 avg). Against a +0 Dex save vs DC 15, the target saves on 15–20 = 30%. Damage distribution: 70% chance of 8d6 (kills at 22+), 30% chance of half-damage (insufficient). Combined kill probability is roughly 50–55% factoring damage variance.

Are online dice rollers actually random?

Reputable rollers like Roll20's Quantum Roll, AnyDice, and Foundry VTT's built-in generator use cryptographically strong pseudo-random generators with periodic reseeding. They are vastly more uniform than cheap injection-molded physical dice.

Why do my 1d20 rolls feel more random than 3d6 rolls?

Because variance is much higher: 1d20 standard deviation is 5.77 versus 3d6 at 2.96. A 1d20 swing of ±10 from average is common; a 3d6 swing of ±6 is two standard deviations and rare (~5%).

How do I calculate damage probabilities for spells like Magic Missile?

Magic Missile (PHB p.257) is 3 darts of 1d4+1 each at level 1, total 3d4+3 = 10.5 avg, min 6, max 15. Each upcast slot adds another dart. Use AnyDice with output 3d4+3 to see the full distribution.

Where can I learn more about probability theory behind D&D dice?

Start with Wikipedia's Dice notation article for the formal definitions, then explore AnyDice tutorials by Jasper Flick. The official Wizards of the Coast D&D 5e rules in the PHB and DMG document every dice mechanic.

Reviewed by: Mustafa Bilgic (Adıyaman, Türkiye), independent operator and tabletop probability researcher. Sources: Wizards of the Coast (D&D 5e PHB/DMG), AnyDice, Wikipedia dice notation. Last updated 2026-05-20.