The rule
Some tabletop games let you roll two dice instead of one and keep only the higher result, called advantage, or only the lower result, called disadvantage. The mechanic was popularised by the fifth edition of Dungeons & Dragons, and similar rules appear in other games under other names. It is a compact way to represent a favourable or unfavourable situation without adding a bonus or penalty to a number.
The attraction is simplicity. There is no arithmetic and no modifier to remember. The cost is that the effect on your chances is not uniform. It depends on how likely success was to begin with.
The core calculation
Suppose a single roll succeeds with probability p. Call the chance of failure 1 − p.
With advantage, you fail only if both dice fail. The two dice are independent, so the chance of failure is (1 − p) × (1 − p), and the chance of success is 1 − (1 − p)². This is a case of complement counting: it is easier to find the chance that neither die succeeds and subtract.
With disadvantage, you succeed only if both dice succeed, so the chance of success is p × p, or p².
Nothing about this depends on the die type, only on the single-roll success chance p.
Worked figures for a twenty-sided die
Take three situations. In each, success on one die is a target number or higher.
Need 16 or more (p = 5/20 = 0.25):
- Advantage: 1 − 0.75² = 1 − 0.5625 = 0.4375, or 43.75%
- Disadvantage: 0.25² = 0.0625, or 6.25%
Need 11 or more (p = 10/20 = 0.5):
- Advantage: 1 − 0.5² = 0.75, or 75%
- Disadvantage: 0.5² = 0.25, or 25%
Need 6 or more (p = 15/20 = 0.75):
- Advantage: 1 − 0.25² = 1 − 0.0625 = 0.9375, or 93.75%
- Disadvantage: 0.75² = 0.5625, or 56.25%
Compare the changes in percentage points:
| Single-roll chance | Advantage | Gain | Disadvantage | Loss |
|---|---|---|---|---|
| 25% | 43.75% | +18.75 | 6.25% | −18.75 |
| 50% | 75% | +25 | 25% | −25 |
| 75% | 93.75% | +18.75 | 56.25% | −18.75 |
The effect is largest when the single-roll chance is 50 percent, where it is worth 25 points either way. It shrinks as the chance moves toward either extreme, because a task that is nearly certain or nearly hopeless has little room to improve or worsen. This is an important difference from a flat bonus of +5, which always moves the chance by 25 points on a twenty-sided die (as long as the target stays within range).
The effect on the average result
Advantage and disadvantage also move the typical number rolled, which is to say the expected value of the roll. A single twenty-sided die averages 10.5. Taking the higher of two averages about 13.8, and taking the lower averages about 7.2. These come from the formula for the expected higher of two N-sided dice, (N + 1)(4N − 1) / (6N), which for N = 20 gives 21 × 79 / 120 = 13.825. The lower of the two is then 21 − 13.825 = 7.175, since the two averages must add to 21.
So advantage is worth roughly +3.3 on average and disadvantage roughly −3.3, which is similar to a modifier of about three. But this is only an average. As the table shows, the equivalent modifier depends on the target, and a fixed +3 would move the chance by 15 points everywhere in range, rather than by 18.75 to 25 points near the middle.
For a six-sided die, the same method gives a higher-of-two average of 161/36, about 4.47, compared with a single-die average of 3.5.
What about natural maximums and minimums
Rules that treat the top face of the die specially, such as an automatic success on a 20, are affected too. With advantage, the chance that at least one die shows a 20 is 1 − (19/20)² = 39/400, or 9.75 percent, nearly double the single-die 5 percent. With disadvantage, you need both dice to show 20, and the chance is 1/400, or 0.25 percent.
Does it fit the situation?
The mechanic works best when it is used sparingly. Because the effect is large near a coin flip and small at the edges, it tends to feel meaningful for even-odds actions and trivial for the nearly certain or nearly impossible. Games commonly address this by letting advantage and disadvantage cancel out, so one of each leaves a normal single roll rather than stacking.
Stacking is a design choice. A rule that allows three dice and keeping the best raises the success chance further: with p = 0.5, the chance becomes 1 − 0.5³ = 87.5%. Whether that is desirable depends on how much the designer wants circumstances to matter.
Reading it correctly
None of this tells you what the next roll will be. Advantage shifts the odds, it does not make any particular die land higher, and a player with advantage can still fail. The calculation also assumes the dice are fair and independent, as covered in the dice probability guide. Over a long campaign, the shifts add up to a steady edge in how often a character succeeds. That is a long-run tendency, in the spirit of the law of large numbers, and short runs of play will vary a great deal because of plain random chance.