Dice Probability: Notation, Advantage Math and Average Damage

Dice notation has the form NdS+M. N is the number of dice, S is the number of sides on each die, and M is a flat number you add or subtract after the roll. So 2d6+3 means: roll two six-sided dice, add them, then add 3. A single die can drop the N, so d20 is the same as 1d20. You can chain terms, for example 1d8+2d6+4.

Three numbers describe any roll:

• Minimum = N + M (every die shows 1).
• Maximum = N × S + M (every die shows its top face).
• Average = N × (S + 1) / 2 + M. One d6 averages 3.5, one d8 averages 4.5, one d20 averages 10.5.

More dice do not change the average per die, but they change the shape. One d12 is flat: each result from 1 to 12 has the same 1-in-12 chance. Two d6 make a bell: of the 36 possible combinations, six add up to 7 (16.7%), but only one makes 2 and only one makes 12 (2.8% each). That is why 2d6 feels more reliable than 1d12.

In D&D 5e, as written in SRD 5.1, advantage means you roll two d20s and use the higher one. Disadvantage means you roll two and use the lower one. Two sources of advantage do not stack: you never roll more than one extra d20. If you have advantage and disadvantage at the same time, they cancel and you roll one d20.

To succeed you need the d20 plus your bonus to equal or exceed the DC (for checks and saves) or the target's AC (for attacks). Call p your chance on one die. With advantage you fail only if both dice fail, so the chance is 1 − (1 − p)². With disadvantage both dice must succeed, so the chance is p².

Each +1 on a d20 is worth 5 percentage points. Advantage is worth the most in the middle: when you need an 11 it adds 25 points, the same as a +5 bonus. At the edges it adds little, only 4.75 points when you need a natural 20. The average of the kept die also moves: 13.825 with advantage and 7.175 with disadvantage, against 10.5 for one d20.

Average damage uses the same formula. A longsword hit in SRD 5.1 deals 1d8 slashing damage, so with a +3 Strength modifier it averages 4.5 + 3 = 7.5. A greatsword deals 2d6, so 2d6+3 averages 7 + 3 = 10. A fireball deals 8d6 fire damage: an average of 28 on a failed save and half of that, 14, on a success.

On a critical hit, SRD 5.1 has you roll the attack's damage dice twice and then add modifiers once. A longsword critical is 2d8+3, which averages 12, not 15.

To compare options, multiply damage by the chance to hit. Take a +5 attack against AC 15. You need 10 or higher on the d20, so you hit 11 times in 20 (55%), and one of those 11 faces is the natural 20 critical. Expected damage per swing with a longsword is 50% × 7.5 + 5% × 12 = 4.35. With advantage the hit chance rises to 79.75% and the critical chance to 9.75%, so the expected damage becomes 70% × 7.5 + 9.75% × 12 = 6.42.