Every d20 roll is a probability problem. Most players develop intuition for it over time — "DC 15 is hard," "advantage is roughly +5" — but the actual math is more interesting than the gut feel. Knowing the numbers won't make you roll better, but it will help you understand why some encounters feel impossible and others feel trivial.
The d20: Uniform Distribution
A d20 gives you a flat 5% chance of landing on any specific number. Every result from 1 to 20 is equally likely. This is a uniform distribution, and it makes the d20 fundamentally different from rolling multiple dice (more on that below).
The practical consequence: to hit a DC (Difficulty Class) of N, you need to roll N or higher (before modifiers). The probability of rolling N or higher on a d20 is:
P(roll ≥ N) = (21 - N) / 20
Some common DCs and their base probabilities (no modifiers):
| DC | Chance to hit | Description | |----|--------------|-------------| | 5 | 80% | Very easy | | 10 | 55% | Easy | | 15 | 30% | Medium | | 20 | 5% | Hard | | 25 | 0% (need mods)| Very hard |
That DC 15 check your DM just called? Without any modifiers, you're failing it 70% of the time. A +5 modifier shifts it to a 55% success rate (you need to roll a 10 or higher). A +8 makes it 70%. Modifiers matter enormously on a flat distribution because each +1 is always worth exactly 5 percentage points.
Advantage and Disadvantage
The 5th Edition mechanic of rolling twice and taking the higher (advantage) or lower (disadvantage) result is one of the most elegant rules in tabletop RPG history. It's also widely misunderstood.
Advantage: Roll 2d20, Take the Higher
The probability of getting at least N with advantage is:
P(max of 2d20 ≥ N) = 1 - P(both dice < N) = 1 - ((N-1)/20)²
Disadvantage: Roll 2d20, Take the Lower
P(min of 2d20 ≥ N) = P(both dice ≥ N) = ((21-N)/20)²
Here's what that looks like for common DCs:
| DC | Normal | Advantage | Disadvantage | |----|--------|-----------|--------------| | 5 | 80% | 96% | 64% | | 8 | 65% | 87.75% | 42.25% | | 10 | 55% | 79.75% | 30.25% | | 11 | 50% | 75% | 25% | | 13 | 40% | 64% | 16% | | 15 | 30% | 51% | 9% | | 18 | 15% | 27.75% | 2.25% | | 20 | 5% | 9.75% | 0.25% |
The "advantage is roughly +5" shorthand comes from looking at the average roll. A straight d20 averages 10.5. With advantage, the average is about 13.82 — a difference of 3.32. But the +5 comparison comes from how advantage affects median outcomes, and it's a rough approximation that breaks down at the extremes. At DC 10, advantage gives you a 24.75 percentage point boost. At DC 20, it only gives you 4.75. Advantage is most powerful in the middle of the range and weakest at the edges.
One critical nuance: advantage and disadvantage don't stack, and they cancel each other out regardless of how many sources of each you have. Three sources of advantage and one source of disadvantage? Straight roll. This is a deliberate design choice to keep the math bounded.
Critical Hits and the Expanded Crit Range
On a standard d20, a natural 20 is a critical hit — a 5% chance per attack. Some class features expand this range:
- Champion Fighter (Improved Critical): Crits on 19-20 = 10% crit chance
- Champion Fighter level 15 (Superior Critical): Crits on 18-20 = 15%
- Hexblade's Curse: Crits on 19-20 = 10%
With advantage, the probability of rolling at least one natural 20 jumps from 5% to 9.75% (nearly doubled). For a Champion with crits on 19-20, advantage pushes the crit chance from 10% to 19% — almost one in five attacks. This is why abilities that grant advantage on attacks (like Reckless Attack, Faerie Fire, or flanking rules) are so powerful for crit-fishing builds.
The exact formula for crit probability with advantage and an expanded crit range (crit on N or higher):
P(crit with advantage) = 1 - ((N-1)/20)²
For crits on 19+: 1 - (18/20)² = 1 - 0.81 = 19%
Multiple Dice: Bell Curves
When you roll multiple dice and add them together (2d6, 3d6, 4d6), the distribution shifts from uniform to a bell curve. The more dice, the tighter the curve around the average.
2d6 (common for damage, or the basis of systems like Traveller and Powered by the Apocalypse):
- Range: 2–12
- Average: 7
- Probability of a 7: 16.67% (6 out of 36 combinations)
- Probability of a 2 or 12: 2.78% each (1 out of 36)
The distribution is triangular — 7 is six times more likely than 2 or 12. This is why 2d6-based systems like PbtA feel fundamentally different from d20 systems. In PbtA, a +1 modifier doesn't add a flat 5% — its effect depends on where you are on the curve. Going from a total of 6 to 7 is a bigger swing than going from 9 to 10.
3d6 (ability score generation, GURPS):
- Range: 3–18
- Average: 10.5
- Probability of a 10 or 11: ~12.5% each
- Probability of a 3 or 18: 0.46% each
This is why rolling 3d6 for ability scores clusters results around 10-11. Getting an 18 is genuinely rare (1 in 216 rolls). The "4d6 drop lowest" method used in 5E character creation skews the curve upward — the average rises to about 12.24, and 18s become a 1.62% event instead of 0.46%.
Damage Dice and Expected Value
Expected value (EV) is the average result over many rolls. For a single die, EV = (max + 1) / 2:
| Die | EV | |-----|------| | d4 | 2.5 | | d6 | 3.5 | | d8 | 4.5 | | d10 | 5.5 | | d12 | 6.5 |
A greatsword (2d6, EV 7) versus a greataxe (1d12, EV 6.5) is a classic comparison. The greatsword has a higher average by 0.5 damage per hit. But the greataxe has a higher ceiling (12 vs 12, same max) and pairs better with features that care about individual die results, like the Barbarian's Brutal Critical (which adds extra weapon dice on a crit — one extra d12 vs one extra d6).
For the greatsword, the Great Weapon Fighting style (reroll 1s and 2s on damage dice) bumps the EV from 7 to 8.33. For the greataxe, it goes from 6.5 to 7.33. The greatsword benefits more from this style because it has two dice that can be rerolled, widening the gap to 1 point of expected damage.
Bounded Accuracy and Why +1 Matters
5th Edition uses a design principle called bounded accuracy: the range of attack bonuses and ACs stays relatively compressed compared to earlier editions. A level 1 character might have a +5 to hit; a level 20 character might have a +11. Enemy ACs range from about 10 (commoner) to 25 (Tiamat).
In this compressed range, each +1 is significant. Against an AC 15 target with a +5 to hit, you need a 10 or higher — 55% hit rate. A +1 weapon bumps that to 60%. A Bless spell (adding 1d4, average +2.5) pushes it to roughly 67.5%. These increments feel small but compound over an adventuring day of dozens of attack rolls.
This is also why AC stacking is so effective. Going from AC 16 to AC 18 (a shield) reduces an enemy's hit chance from, say, 50% to 40% — a 20% relative reduction in hits taken. That's a bigger defensive upgrade than most hit point increases at the same level.
Roll the Numbers
Understanding the math doesn't replace the drama of a clutch natural 20 or the despair of rolling a 1 when it matters most. But it does help you build characters that lean into probability rather than fighting it, and it helps DMs set DCs that create the tension they want at the table.
Our dice roller supports d4 through d100 and lets you roll up to 10 dice at once — useful for checking out damage distributions or just settling an argument about how 4d6-drop-lowest actually shakes out.