Dice probability: the odds for 2d6, 3d6 and d20

Updated 17 October 2026

All the figures in this guide are exact. They come from counting every possible way the dice can land, which is the whole trick behind dice probability: list the equally likely outcomes, count the ones you care about, and divide.

One die: flat odds

A fair die with n sides gives each face a 1 in n chance: 1 in 6 on a d6, 5% on a d20, 1% on a d100. Because every face is equally likely, the average roll is the midpoint between the lowest and highest face: 3.5 on a d6, 10.5 on a d20. You can never actually roll the average on those dice, but over many rolls the results settle around it.

Two dice: why 7 wins

Roll two six-sided dice and there are 6 × 6 = 36 combinations, all equally likely, if you tell the dice apart (a red 1 and a blue 6 is not the same as a red 6 and a blue 1). The totals are not equally likely, because they can be made in different numbers of ways.

7 is special because every first roll can still make it: 1 needs a 6, 2 needs a 5, and so on up to 6 needing a 1. For any other total, some first rolls already rule it out. A first roll of 1 can’t make 8 or more, which is why the counts fall away symmetrically on both sides of 7.

A few useful facts that follow from the table:

• 6, 7 or 8 comes up 16 times in 36, or 44.4% of rolls, nearly half.
• Doubles (6 of the 36 combinations) also come up 1 roll in 6.
• Needing 8 or more is a 41.7% shot; needing 10 or more only 16.7%, or 1 in 6.

Three dice: a bell curve

With three six-sided dice there are 216 combinations, and the middle bulges further. 10 and 11 are the most common totals, with 27 ways each (12.5%). A 3 or an 18 needs all three dice to agree, which happens once in 216 rolls (0.46%). Nearly half of all 3d6 rolls (48.1%) land between 9 and 12.

The more dice you add, the more the totals bunch up around the average, and the rarer the extremes become. That is why game designers reach for several dice when they want reliable, middling results and a single die when they want anything to be possible.

The d20, advantage and disadvantage

A single d20 is flat: needing 11 or more is a 50% chance, needing 15 or more is 30%. In Dungeons & Dragons, advantage means rolling two d20s and keeping the higher, and disadvantage means keeping the lower. You only fail with advantage if both dice fail, so the chance of success is 1 minus the chance of failing, squared.

Advantage is worth the most for middling targets: at 11+ it adds 25 percentage points. For very easy or very hard rolls it adds much less, under 5 points at 2+ or 20+. On average, advantage turns the d20’s 10.5 into 13.825, and disadvantage drags it down to 7.175.

4d6, drop the lowest

The rolling method for D&D ability scores is to roll four d6s and add up the highest three. Dropping the lowest die pulls results upwards: the average is about 12.24 instead of 10.5 for a plain 3d6. The most common score is 13 (13.3%), an 18 turns up in 1.6% of rolls, and a 3 needs four 1s, a 1 in 1,296 chance. The alternative standard array (15, 14, 13, 12, 10 and 8) averages exactly 12, so rolling gives you slightly more on average, at the cost of a wider spread.

The bet that started probability theory

In 1654 the French writer Antoine Gombaud, known as the Chevalier de Méré, brought a dice puzzle to Blaise Pascal, and the letters Pascal and Pierre de Fermat exchanged about such problems are often called the birth of probability theory. The puzzle is easy to state. At even money, betting on at least one six in four rolls of a die was a winning bet. Why, then, did betting on at least one double six in 24 rolls of two dice seem to lose, when 24 is to 36 as 4 is to 6?

The answer is to count the chance of never succeeding. Four rolls with no six: (5/6)⁴ = 48.2%, so at least one six is 51.8%. Twenty-four rolls with no double six: (35/36)²⁴ = 50.9%, so at least one double six is only 49.1%. Chances of “at least once” don’t grow in proportion to the number of tries, and that small gap is the difference between a winning and a losing bet.

Dice have no memory

If 7 hasn’t come up for twenty rolls, it is not “due”. Each roll is independent, so the next one has exactly the same 1 in 6 chance. Long dry spells are common: the chance of going 10 rolls of 2d6 without a single 7 is (5/6)¹⁰, about 16%. What evens out over thousands of rolls is the proportion, not the count.

You can try it yourself with the dice roller, which rolls up to ten dice of any standard type and adds them up. Its rolls come from your browser’s secure random number generator; read how random a random number generator is for why that makes every face exactly equally likely.