Advantage and Disadvantage in Dice Games: What Are the Real Odds?
Advantage selects the higher of two independent d20 results, while disadvantage selects the lower of two independent d20 results.
A worked example with numbers
For a target of 15, one d20 succeeds on 15, 16, 17, 18, 19, or 20: 6 faces of 20, so p = 0.30. Advantage fails only if both dice fail, which has probability 0.70 × 0.70 = 0.49; therefore success is 1 − 0.49 = 0.51, or 51%. Disadvantage succeeds only if both dice succeed: 0.30 × 0.30 = 0.09, or 9%. The same target therefore does not receive a fixed numerical bonus.
How to set the rule before the result
Use the advantage and disadvantage roller to display both d20 values and the kept result. Begin by converting a game target and modifier into the minimum natural d20 face required. If a +4 modifier must meet difficulty 17, the die must show 13 or more, so p = 8/20 = 40%; advantage gives 1 − 0.60² = 64%, and disadvantage gives 0.40² = 16%. Apply any game-specific natural-1 or natural-20 exception separately.
Common mistakes that change the odds or the process
Do not call advantage “+5” as an exact rule. At a 50% base chance, advantage increases success to 75%, but near 0% or 100% its effect is smaller because probabilities cannot leave that range. Do not add two advantage sources into three dice unless the game explicitly says so. Do not take the two dice’s sum or average: the rule is to keep one die, and the kept-value distribution is different.
Where this method stops being appropriate
These equations require two independent, uniformly distributed d20 rolls and a pass/fail target based only on the kept die plus a fixed modifier. They do not automatically cover critical rules, rerolls, dice substitution, multiple targets, or systems that use the words advantage and disadvantage differently. Read the particular game’s rule before applying a D&D-style formula to another dice game.
How the random source fits into the rule
the D&D Beyond Basic Rules states that advantage rolls a second d20 and uses the higher result, while disadvantage uses the lower result. The formula follows from the probability that both rolls fail or both rolls succeed. A browser can choose digital faces with the method in MDN’s Crypto.getRandomValues reference, but the game rule—not the page animation—defines which result is kept.
See the entire d20 threshold pattern
| Minimum natural d20 | Normal | Advantage | Disadvantage |
|---|---|---|---|
| 6 | 75% | 93.75% | 56.25% |
| 11 | 50% | 75% | 25% |
| 16 | 25% | 43.75% | 6.25% |
At each row, advantage is 1 minus the square of failure, while disadvantage is the square of success.
Why the kept result changes the average
One fair d20 has average 10.5. With advantage, high pairs such as (18,4) keep 18 while low pairs retain only when both dice are low; the average kept d20 is 13.825. Disadvantage is the mirror image with average 7.175. These averages describe the kept die before modifiers. They do not say that a particular target receives an equivalent flat +3.325 or −3.325 bonus.
Work from a modifier to a needed face
For difficulty 18 with a +5 modifier, success needs natural 13 or higher: 8 faces out of 20, p = 40%. Advantage produces 1 − 0.60² = 64%; disadvantage produces 0.40² = 16%. A modifier changes the threshold first, then the two-roll formula applies. If the threshold is 1 or lower, ordinary success is already 100% unless the game adds a special failure rule.
Keep rule exceptions outside the base formula
Some systems make a natural 20 automatically successful or a natural 1 automatically unsuccessful for particular roll types. Those rules alter the set of successful faces; apply them before using p. Other games use advantage terminology for extra dice, rerolls, or stacked effects. Quote the game’s text and calculate its specific rule rather than importing a d20 formula by name alone.
Compare outcomes, not a slogan
At a required natural 20, normal success is 5%, advantage is 1 − 0.95² = 9.75%, and disadvantage is 0.05² = 0.25%. At a required natural 2, normal success is 95%, advantage is 99.75%, and disadvantage is 90.25%. These endpoints show why “advantage equals a fixed bonus” fails: its percentage effect varies with the target.
For a game session, report the needed face, modifier, and any exception alongside the percentage. A player can then reproduce the result instead of relying on an unexplained label such as “good odds.”
Count the 400 ordered pairs
Two d20 rolls produce 20 × 20 = 400 ordered pairs. For a minimum 15, advantage succeeds in every pair except the 14 × 14 = 196 pairs where both dice are below 15; 204/400 is 51%. Disadvantage succeeds only in the 6 × 6 = 36 pairs where both dice meet 15; 36/400 is 9%.
The pair count also explains ties between the two raw dice: (15,15) is one successful advantage pair and one successful disadvantage pair because both values are identical. No special tie-break is needed; the kept value is 15 in either mode.