Advantage and Disadvantage Rolls Explained

The rule in tabletop games

Some tabletop role-playing games use a mechanic where you roll a twenty-sided die (a d20) twice. With advantage you keep the higher result. With disadvantage you keep the lower. The change feels large at the table, but the exact size of the effect depends on how hard the target number is. The maths is short and uses only ideas from basic probability.

One roll first

Suppose you need to roll a target number T or higher on a d20. The faces that succeed are T, T+1, … up to 20, which is 21 - T faces out of 20. So the single-roll chance is

p = (21 - T) / 20

A target of 11 gives p = 10/20 = 0.50. A target of 15 gives p = 6/20 = 0.30.

Advantage

You succeed with advantage if at least one of the two dice reaches the target. It is easier to count the opposite: you fail only if both dice miss. Each die misses with chance 1 - p, and the dice are independent, so both miss with chance (1 - p) squared. This is the complement rule at work:

advantage chance = 1 - (1 - p)^2

For a target of 11, p = 0.5, so the advantage chance is 1 - 0.25 = 0.75. That is a rise from 50 percent to 75 percent.

Disadvantage

With disadvantage the lower die is kept, so you succeed only if both dice reach the target. That is p x p:

disadvantage chance = p^2

For a target of 11, that is 0.5 x 0.5 = 0.25, which is 25 percent.

A table of chances

These values are computed from the formulas above. Percentages are rounded.

Check one row by hand: for a target of 15, p = 0.3 and 1 - p = 0.7. Squaring gives 0.49, so advantage is 1 - 0.49 = 0.51. Disadvantage is 0.3 x 0.3 = 0.09.

What the table shows

Advantage is not a flat bonus. In the middle of the range it helps the most. At a target of 11 it adds 25 percentage points. At a target of 5, where success is already likely, it adds 16 points, and at a target of 20 it adds only 4.75 points. The same pattern appears for disadvantage, which costs the most when the single-roll chance is near 50 percent and costs less when success is already unlikely or very likely.

Some players describe advantage as roughly a “+5” to the roll. That is a handy rule of thumb in the middle of the range, since each point of target on a d20 is worth 5 percentage points of a single roll. A shift from 50 to 75 percent equals about five points. But it is an approximation and it breaks down near the extremes, where the true gain is smaller.

Natural 20s

The same formulas apply to rolling a 20 itself. With a single die, p = 1/20 = 5 percent. With advantage, the chance of at least one 20 is 1 - (19/20)^2 = 1 - 361/400 = 39/400 = 9.75 percent. With disadvantage, you need both dice to show 20, which is 1/400 = 0.25 percent. If a game gives special effects to a natural 20 or a natural 1, advantage and disadvantage change how often they occur.

Does the average roll change?

The average of a single d20 is 10.5. Taking the higher of two d20s gives an average of 13.825, and taking the lower gives 7.175. These follow from counting the 400 equally likely pairs, and they are symmetric around 10.5, so the shift is about 3.3 up or down. Those averages describe the long-run mean of many rolls. Any single roll can still be anything from 1 to 20, because the dice remain random. The flat spread of one d20 changes to a skewed one, with higher faces more likely under advantage.

Practical points

• Results of separate rolls are independent. A failed roll earlier does not make the next one more likely to succeed, a point tied to the gambler’s fallacy.
• If a game allows advantage and disadvantage together, many rule sets say they cancel out, however many of each there are. That is a rule of the game, not a result of probability.
• Rolling three dice and keeping the highest raises the chance further: 1 - (1 - p)^3. At a target of 11 that is 1 - 0.125 = 0.875.

These calculations describe a fair die. They explain how a game mechanic behaves; they do not suggest any approach to betting. Chance involves variance, and a good probability does not guarantee a good outcome in any one roll.