Quick Answer: A d20 has a flat 5% probability for each result from 1-20. With a +5 modifier, you need to roll 10+ to hit AC 15 (60% success). With Advantage (roll 2d20, take higher), your probability of rolling 15+ increases from 30% to 51%. Critical hits (natural 20) always occur 5% of the time.
The Flat d20 Distribution
Unlike 2d6 (which creates a bell curve), a single d20 produces a perfectly flat distribution — every number from 1 to 20 has exactly 5% probability. This means success and failure are determined almost entirely by your modifier, not by variance favoring middle results.
Modifiers and Success Probability
| Target Number (DC/AC) | Modifier +0 | Modifier +3 | Modifier +5 | Modifier +8 |
|---|---|---|---|---|
| 10 | 55% | 70% | 80% | 95% |
| 12 | 45% | 60% | 70% | 85% |
| 15 | 30% | 45% | 55% | 70% |
| 18 | 15% | 30% | 40% | 55% |
| 20 | 5% | 15% | 25% | 45% |
Advantage and Disadvantage
Advantage: Roll 2d20, take the higher result. This is not just an average shift — it changes shape. P(rolling X or higher with Advantage) = 1 - P(both dice roll lower than X) = 1 - ((X-1)/20)². With Advantage, hitting a DC 15 increases from 30% to 51%.
Disadvantage: Roll 2d20, take the lower result. P(rolling X or higher with Disadvantage) = (P(natural roll ≥ X))² = ((21-X)/20)². With Disadvantage, hitting a DC 15 drops from 30% to 9%.
Critical Hits: 5% Every Time
A natural 20 (d20 shows 20) is always 5% — regardless of modifier or context. This cannot be improved by modifiers (though features like Champion Fighter in D&D 5e lower the critical hit threshold to 19-20, making it 10%). Critical hits double the weapon's damage dice.