D&D Dice Notation and Probability Explained

Two six-sided dice can land in 36 different ways, and six of them add up to 7. That single fact explains a lot about tabletop dice. Once you can read a code like 2d6+3, you can find its average and its odds in under a minute.

What Does a Code Like 3d6+2 Actually Mean?

The code 3d6+2 tells you to roll three six-sided dice, add the faces, and then add 2. The first number is how many dice you roll. The number after the “d” is how many faces each die has.

Players often shorten 1d20 to just d20, since one die is the default. The number after the plus sign is a modifier. It gets added once to the total, not once per die.

A minus sign works the same way in reverse. So 1d20-1 means roll one d20 and subtract 1 from the result. Tables sometimes chain codes together too, such as 2d6+1d8 for a weapon with bonus damage.

The Standard Dice Set

Tabletop games use six die shapes: d4, d6, d8, d10, d12, and d20. Two d10s rolled together, one read as tens, give a number from 1 to 100, written d100 or d%. The d20 does most of the work, since it decides attacks, checks, and saving throws.

The notation also allows extra instructions. For example, 4d6 drop lowest means you roll four dice and add only the best three.

Averaging Any Roll in Two Steps

Every fair die has faces from 1 up to its largest number, and each face is equally likely. So the average of one die sits halfway between 1 and the top face. The formula is short: average of one die = (X + 1) / 2.

For a full code, use two steps:

• Multiply the one-die average by the number of dice, N.
• Add the modifier once at the end.

This works because the average of a sum equals the sum of the averages. So 3d6+2 averages 3 x 3.5 + 2, which is 12.5. A spell that deals 8d6 averages 8 x 3.5, or 28 damage, even though it can roll anywhere from 8 to 48.

An average is a long-run value, not a promise for one roll. Our guide to how expected value works covers that idea in more depth.

A Longsword Attack From Roll to Damage

Here is the example from our calculator page. A level 5 fighter makes 2 attacks per round with a +7 attack bonus. The target has an armor class, or AC, of 16. Each hit deals 1d8+4 with a longsword.

Step 1: The Hit Chance

The fighter needs 16 – 7 = 9 or higher on the d20. That covers the faces 9 through 20, which is 12 of 20 faces. So each swing hits 60 percent of the time.

Step 2: Normal Hits and Critical Hits

A natural 20 always hits and counts as a critical hit, so crits happen 5 percent of the time. On a crit, you roll the damage dice twice and add the modifier once. That leaves 55 percent normal hits worth 8.5 on average and 5 percent crits worth 2 x 4.5 + 4 = 13.

Step 3: Damage per Round

One swing averages 0.55 x 8.5 + 0.05 x 13, which equals 5.325 damage. Two swings give 10.65, or about 10.7 damage per round. Each hit that lands averages about 8.9 damage once crits are included.

Why Do Bigger Dice Pools Cluster Near the Middle?

Middle totals have more dice combinations behind them than the extremes do. A 2 on 2d6 needs both dice to show 1, but a 7 can come from six different pairs.

Treat the two dice as a first die and a second die. Each has 6 faces, so there are 6 x 6 = 36 equally likely pairs. The pairs 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1 all make 7, so 7 has a chance of 6/36, or 16.7 percent.

The effect grows with more dice. On 3d6, the totals 10 and 11 each come up 27 times in 216, which is 12.5 percent. An 18 needs three sixes, a chance of 1 in 216.

This shape matters at the table. A 2d6 weapon and a 1d12 weapon share the same top damage of 12. The 2d6 weapon averages 7 against 6.5, and it lands between 6 and 8 on 44 percent of rolls. A single d20 has no middle bump at all, since each face stays a flat 5 percent.

Where Dice Math Trips Players Up

Most dice errors come from skipping a rule or averaging the wrong way. These five show up at many tables.

• Adding the modifier to every die. The code 3d6+2 adds 2 once, for an average of 12.5. Adding it to each die gives 16.5, which is wrong.
• Doubling the modifier on a crit. A critical hit doubles the dice only. So 1d8+4 becomes 2d8+4 on a crit, which averages 13, not 17.
• Treating 2d6 as flat. A 12 on 2d6 comes up 1 time in 36, not 1 time in 11. Only a single die gives every number the same chance.
• Adding chances across rounds. A 5 percent crit chance does not reach 100 percent after 20 swings. The odds of at least one success follow a different rule, which our guide to how drop chances stack over many tries explains.
• Forgetting the automatic results. A natural 20 hits even a very high AC, and a natural 1 misses even a low one.

A good habit is to say the average out loud before you roll. It helps you spot a wrong total in seconds.

Is Advantage Worth More Than a Flat Bonus?

Near the middle of the die, yes. At a target of 11, advantage lifts your chance from 50 to 75 percent, the same jump as a +5 bonus. Near the edges, it adds much less.

With advantage, you roll a second d20 and use the higher roll. With disadvantage, you use the lower roll. With advantage, you fail only when both dice fail. So your success chance equals 1 minus the fail chance squared.

Run the fighter from earlier with advantage. The hit chance rises from 60 to 84 percent, and the crit chance nearly doubles to 9.75 percent. Damage per round climbs from 10.65 to about 15.2. A flat +2 bonus, by comparison, raises it only to about 12.4.

Disadvantage works the other way and hurts just as much. The average roll drops from 10.5 to 7.175, and the 60 percent hit chance falls to 36 percent. To compare these options for your own build, enter each version in our damage per round calculator and read the result side by side.

Common Questions About D&D Dice

References

This article is general math and gaming education. Probabilities assume fair dice and the attack rules in the free system reference document. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.