D&D 5e Advantage & Disadvantage Math: The Exact Probabilities
Deconstructing the most iconic mechanic of 5th Edition Dungeons & Dragons through discrete order statistics.
While the Dungeon Master's Guide frequently uses +5 as a passive bonus equivalent for Advantage (such as in Passive Perception), the true mathematical value varies depending on the target number needed on the d20:
- Maximum Benefit (+5.0): When you need an 11 to hit (a 50% baseline chance), Advantage increases your success rate to 75.0% (+25% absolute gain).
- Minimum Benefit (+0.95): When you need a Natural 20 (5% baseline), Advantage increases your chance to 9.75% (+4.75% absolute gain, equivalent to a +0.95 bonus).
- Critical Hit Chance: Jumps from 5.0% to 9.75% (almost 1 in 10 attacks).
Complete d20 Advantage & Disadvantage Probability Matrix
The table below displays the probability of rolling at least the target number on a d20 under all three conditions:
| Roll Needed | Normal (1d20) | With Advantage | With Disadvantage | Effective Bonus |
|---|---|---|---|---|
| 2+ | 95.0% | 99.75% | 90.25% | +0.95 |
| 4+ | 85.0% | 97.75% | 72.25% | +2.55 |
| 6+ | 75.0% | 93.75% | 56.25% | +3.75 |
| 8+ | 65.0% | 87.75% | 42.25% | +4.55 |
| 11+ (50%) | 50.0% | 75.00% | 25.00% | +5.00 (Max) |
| 14+ | 35.0% | 57.75% | 12.25% | +4.55 |
| 16+ | 25.0% | 43.75% | 6.25% | +3.75 |
| 18+ | 15.0% | 27.75% | 2.25% | +2.55 |
| 20 (Nat 20) | 5.0% | 9.75% | 0.25% | +0.95 |
Why the Effective Bonus Changes
The reason Advantage feels so powerful during gameplay is because most tabletop combat encounters are calibrated around an attack roll landing on an 8 through 14. Within this critical window, Advantage provides an effective numerical bonus between +4.5 and +5.0.
However, when you need an extreme roll (such as needing an 18, 19, or 20 to hit an ancient dragon with 22 AC), Advantage cannot conjure bonuses out of thin air. You still only have two dice, meaning your chances of hitting an impossible target only rise moderately.
Elven Accuracy (Triple Advantage)
The Elven Accuracy feat allows Elves and Half-Elves to reroll one of their dice whenever they make an attack using Dexterity, Intelligence, Wisdom, or Charisma with Advantage. Mathematically, this equals rolling 3d20 and keeping the highest:
- Standard Hit Rate (Target 11+): Jumps from 50.0% to 87.5%!
- Natural 20 Crit Rate: 1 - (19/20)³ = 14.26% (roughly 1 in every 7 attacks).
- Champion Fighter Crit Rate (19-20): 1 - (18/20)³ = 27.10% (more than 1 in every 4 swings is a critical hit).