What a dice probability actually is
Every dice question reduces to counting. A fair die numbered 1 to X gives each face a probability of 1/X, and rolling several dice and adding them produces a distribution over totals that is no longer flat: there is exactly one way to roll 2 on 2d6 and six ways to roll 7, so a 7 is six times as likely. The set of all those probabilities is the sum distribution, and once you have it every other question - beat a DC, roll in a range, average damage - is arithmetic on that one object.
The distinction that matters most at the table is between a single die and a pool. One d20 is uniform: every result from 1 to 20 is equally likely, so the chance of beating a target rises in flat 5% steps and a +1 bonus is worth exactly 5 percentage points at any DC in range. A pool of dice is not uniform: 3d6 clusters hard around 10 and 11, so a +1 there is worth much more near the middle of the range than at the edges. Systems built on d20 rolls feel swingy for exactly this reason, and systems built on dice pools feel predictable.
The second thing worth knowing is how spread behaves as you add dice. The mean of NdX is N(X+1)/2 and the variance is N(X2-1)/12, so the mean grows in proportion to N while the standard deviation grows only with the square root of N. Roll four times as many dice and the average quadruples but the spread only doubles, which is why 8d6 fireball damage is far more reliable in relative terms than 2d6.
The three formulas doing the work
Beating a target with one die. If you need t or higher on a dX, the faces that succeed are t, t+1, ..., X, which is X - t + 1 faces out of X. On a d20 needing 15, that is 20 - 15 + 1 = 6 faces, so 30%. Subtract your modifier from the DC first: with +5 against DC 15 you need a 10 on the die, so 11 faces, 55%.
Beating a target with several dice. There is no shortcut worth memorising, so build the distribution. Start with one die - a flat 1/X on each face - then for each additional die, spread every existing total across X new totals. That operation is convolution, and it enumerates every combination without listing them. This calculator does it exactly, in rational arithmetic scaled to floating point, so a 3d6 result of 1/216 comes out as 0.462963% rather than a simulated approximation.
Advantage and disadvantage. These are the cleanest identities in the game. Rolling twice and keeping the better result fails only when both rolls fail, so the failure probability is (1 - p)2 and the success probability is 1 - (1 - p)2. Disadvantage succeeds only when both succeed, so it is p2. Neither identity cares what the underlying roll is, which is why they apply equally to a d20 check and to a whole pool rolled twice.
A consequence worth internalising: advantage is worth most when the roll is closest to a coin flip. The gain 1 - (1-p)2 - p = p(1 - p) is largest at p = 0.5, where it adds 25 percentage points. At a 95% roll it adds 4.75 points and at a 5% roll it adds the same 4.75. In expected-value terms advantage on a d20 raises the average roll from 10.5 to 13.825, a gain of 3.325 - roughly a +3 bonus, but concentrated where it matters rather than spread evenly.
Worked example: a +5 attack against AC 15, with and without advantage
Your attack bonus is +5 and the target's AC is 15. In D&D 5e a roll that equals the AC hits.
- Reduce to the die. You need a total of 15, and the modifier supplies 5, so the die must show 15 - 5 = 10 or higher.
- Count faces. The faces 10 through 20 succeed: 20 - 10 + 1 = 11 faces out of 20, so p = 11/20 = 55%.
- Advantage. Failure needs both dice to show 9 or less: (1 - 0.55)2 = 0.452 = 0.2025. So success is 1 - 0.2025 = 79.75%.
- Disadvantage. Both dice must succeed: 0.552 = 30.25%.
- The size of the swing. Advantage adds 79.75 - 55 = 24.75 percentage points; disadvantage costs 24.75. At a 55% roll you are almost exactly at the peak of the advantage curve.
- Compare with a flat bonus. A +1 item would take you from 55% to 60%, five points. Advantage is worth nearly five such items here - but at a DC where you already succeed 90% of the time, advantage is worth only 9 points, less than two.
- Convert to damage. If a hit deals 2d6+3, the average hit is 7 + 3 = 10, so your expected damage per attack is 0.55 x 10 = 5.5 on a straight roll and 0.7975 x 10 = 7.975 with advantage, before critical hits.
Those last two lines are the reason to compute rather than estimate. Advantage in this spot is a 45% increase in expected damage output, which no amount of table intuition reliably delivers.
Reading the result at the table
Treat 55-65% as the healthy band for a routine attack or check. Below about 35% the action is a gamble that will usually fail, and a character with a better bonus or a different approach should be taking it. Above about 85% the roll is nearly automatic and, in most systems, a good DM should consider not calling for it at all.
Read the standard deviation as the width of a typical miss. On 1d20 the standard deviation is 5.77, which is larger than most character bonuses - a fact that explains why a +2 difference in modifiers feels invisible at the table. On 3d6 the standard deviation drops to 2.96 on a range that is nearly as wide, which is why systems using 3d6 for resolution make character bonuses feel decisive.
When comparing two options, compare expected values, not probabilities. An attack that hits 55% of the time for 10 damage beats one that hits 75% of the time for 6, because 5.5 exceeds 4.5. The DPS calculator formalises this for damage-over-time comparisons, and the same hit-chance-times-damage structure applies whether the numbers come from a video game or a character sheet.
Finally, remember what the probability does not include. Critical hits, the natural-20 and natural-1 rules, reroll abilities such as Lucky and Halfling Luck, and features that let you swap dice after seeing them all sit outside this model. Rerolls and swaps only ever move your true chance up. The natural-20 and natural-1 rules move it both ways: on a 5e attack roll a 20 always hits, so a target this calculator prices at 0% still lands 5% of the time, and a natural 1 always misses, so a roll it prices at 100% is really 95%. Both corrections bite only at the two extremes, and nowhere in between.
d20 target numbers with advantage and disadvantage
| Need on the die | Straight roll | Advantage | Disadvantage | Points advantage adds |
|---|---|---|---|---|
| 2+ | 95.00% | 99.75% | 90.25% | +4.75 |
| 4+ | 85.00% | 97.75% | 72.25% | +12.75 |
| 6+ | 75.00% | 93.75% | 56.25% | +18.75 |
| 8+ | 65.00% | 87.75% | 42.25% | +22.75 |
| 10+ | 55.00% | 79.75% | 30.25% | +24.75 |
| 11+ | 50.00% | 75.00% | 25.00% | +25.00 |
| 12+ | 45.00% | 69.75% | 20.25% | +24.75 |
| 14+ | 35.00% | 57.75% | 12.25% | +22.75 |
| 16+ | 25.00% | 43.75% | 6.25% | +18.75 |
| 18+ | 15.00% | 27.75% | 2.25% | +12.75 |
| 20 only | 5.00% | 9.75% | 0.25% | +4.75 |
The final column is p(1 - p) expressed in percentage points, which is why it peaks at a 50% roll and shrinks symmetrically toward both extremes. Advantage raises the average d20 result from 10.5 to 13.825.
Mistakes and assumptions to watch
- Forgetting that meeting the DC succeeds. In D&D 5e a total equal to the DC is a success, so DC 15 needs a 15, not a 16. Systems differ; check yours before trusting any table.
- Adding advantage to a dice pool. This calculator rolls the entire pool twice and keeps the better total, which is a coherent mechanic but not 5e's. In 5e advantage applies only to a single d20 roll and never to damage dice.
- Treating advantage as a flat +5. It averages +3.325 on a d20 and its effect on success chance ranges from under 5 points to 25, depending entirely on the target number.
- Assuming multiple sources of advantage stack. In 5e they do not: two sources of advantage still give one extra die, and one advantage plus one disadvantage cancels to a straight roll no matter how many of each apply.
- Ignoring critical hits and auto-success rules. A natural 20 always hits and a natural 1 always misses on 5e attack rolls, independent of the arithmetic. That floor and ceiling matter most when the calculated chance is near 0% or 100%.
- Assuming dice are fair. Cheap injection-moulded dice are measurably biased, and digital rollers depend on the quality of their generator. The model here assumes a perfectly fair die, which is a good approximation for casino-quality dice and an optimistic one for a bag of dollar-store d20s.
Why 3d6 and 1d20 feel completely different
Both span roughly the same range - 3 to 18 against 1 to 20 - and both average about 10.5. The difference is entirely in the spread. On 1d20 every result is equally likely, so a roll of 3 is exactly as common as a roll of 11. On 3d6 the result 10 or 11 comes up 12.5% of the time each while a 3 comes up 0.46% of the time, and 67.6% of all rolls land between 8 and 13. That is why swapping 3d6 for 1d20 in an otherwise unchanged system makes character bonuses dominate outcomes: the dice stop generating surprises, and the modifier becomes the story.
Related dice and probability tools
Dice questions branch into three families. Sum questions - what does NdX total - are what this calculator handles by convolution. Success-counting pools, used by World of Darkness, Shadowrun and Genesys, ask how many dice beat a threshold rather than what they sum to, and that is a binomial question: n dice each succeeding with probability p. Keep-highest mechanics, such as rolling 4d6 and dropping the lowest for ability scores, need order statistics; the advantage identity here is the simplest case of exactly that.
For repeated attempts at a rare outcome - hunting a specific roll over many sessions, or a loot table - the right tool is the drop rate probability calculator, which handles "at least one in N tries" and the confidence level you need before giving up. For encounter design where the question is how much the dice have to deliver, the encounter XP budget calculator sizes the fight, and the effective health pool calculator converts armour and resistance into the survivability those rolls are chewing through.
One caution about simulators. Rolling ten thousand virtual dice gives an answer accurate to roughly one percentage point, which is fine for a feel but useless for comparing two options that differ by half a point. Convolution gives the exact answer at the same cost, so there is no reason to accept simulation noise for a question this small. Where simulation genuinely earns its place is in sequential problems - a whole combat with conditions, reactions and healing - because those have too many interacting states to enumerate.
Key terms
- NdX
- Standard dice notation: roll N dice with X faces each and add the results. 3d6 means three six-sided dice, giving totals from 3 to 18.
- Convolution
- The operation that combines two distributions into the distribution of their sum. Adding a die to a pool means spreading each existing total across X new totals, each with probability 1/X.
- Standard deviation
- The typical distance of a roll from its average. On 1d20 it is 5.77; on 3d6 it is 2.96, which is why pooled dice feel so much more predictable.
- Advantage
- Roll twice, keep the better result. Its success probability is 1 - (1 - p)^2, and its benefit peaks at 25 percentage points when the straight roll is exactly 50%.