Average roll of a D20 and advantage odds

D20 probability calculator with modifiers

Enter the total you need and your modifier, then compare one die with keeping the highest or lowest of several. Two dice match the numerical D&D advantage/disadvantage calculation; larger counts describe a separate keep-one rule.

What is the average roll of a D20?

Add the twenty possible face values and divide by twenty: 210 / 20 = 10.5. This is a long-run average, not a possible face on a single die.

Average D20 roll with advantage and disadvantage

The average roll of a D20 with advantage is 13.825: roll two independent dice and keep the higher. Disadvantage keeps the lower and averages 7.175. These are averages of the kept face before modifiers, not sums of both dice. Neither mode guarantees a particular result.

There are 400 equally likely ordered pairs of two D20s. Exactly 2k − 1 pairs have a highest face of k, because k² − (k − 1)² = 2k − 1. Weight each face by its count: adding k × (2k − 1) for k = 1 through 20 and dividing by 400 gives 13.825. For the lowest face, the count is 41 − 2k; the same weighted calculation gives 7.175.

For example, a highest face of 20 appears in 39 of the 400 pairs, so advantage gives a 9.75% chance of a natural 20. Two standard dice average 21 when added together, but that is a different operation from keeping one die.

Chance of meeting a target

For an integer target t from 1 to 20, standard success is (21 − t) / 20. With advantage it is 1 − ((t − 1) / 20)². With disadvantage it is ((21 − t) / 20)².

Modifiers and limits

Subtract your modifier from the desired total to obtain the required face. A required face at or below 1 is certain; above 20 is impossible. These are mathematical thresholds, not automatic-success rules for a particular game.

A fixed +2 modifier raises the standard average total to 12.5; +5 raises it to 15.5. The average raw face stays 10.5. Try rolling a D20 with advantage or read how D20 rolls and modifiers work.

Advantage versus a flat bonus

Advantage adds 3.325 to the average kept face, but it is not equivalent to a fixed +3 or +5 bonus at every target. If you need an unmodified 11, standard succeeds 50% of the time and advantage 75%: a gain of 25 percentage points. At target 20, those chances are 5% and 9.75%, a gain of 4.75 points.

For target 15, standard is 30%; advantage is 51%; standard with +5 is 55%. A flat modifier can also reach totals above 20, while keeping a higher unmodified D20 cannot. These comparisons ignore a game’s automatic-hit or failure rules.

Repeated rolls and natural 20s

On n independent standard rolls, the chance of at least one natural 20 is 1 − (19/20)ⁿ. Six rolls give about 26.491%; 100 rolls give about 99.408%. That is different from the chance that every roll is a 20, which is (1/20)ⁿ.

A batch of 100 fair D20s has an expected five natural 20s and an expected raw sum of 1,050. An expectation is not a quota or guarantee. No 20s in that batch still has about a 0.592% chance, and a previous low roll does not change the next die’s 5% chance.

For the sum of n standard D20s, the average is 10.5 × n. The average of 2d20 added together is 21, compared with 13.825 when keeping the higher of two. Use a batch of D20 rolls to compare a sample with these expectations; one small sample cannot prove or disprove fairness.