Dice probability guide

The basic rule: count the outcomes

A fair die has the same chance of landing on each face. If a die has S sides, the probability of any particular face is 1 ÷ S. For an event made of several faces, count the faces that match and divide by the total:

Probability = number of outcomes you want ÷ number of possible outcomes

Example: the chance of rolling 5 or 6 on a six-sided die is 2 ÷ 6 = 1/3, or about 33.33%.

When you roll several dice, multiply the number of sides to get the number of possible outcomes. Two six-sided dice have 6 × 6 = 36 outcomes, three have 6 × 6 × 6 = 216, and five have 6⁵ = 7,776. Each of these outcomes is equally likely, so the job is always to count the ones you are interested in.

One die of each type

With a single die every face is equally likely, and the average result is halfway between the lowest and highest face: (1 + S) ÷ 2.

Two six-sided dice (2d6)

Adding two dice changes the picture completely. The 36 outcomes are not spread evenly over the totals 2 to 12. A total of 2 needs both dice to show 1, which happens in only one way. A total of 7 can be made in six ways: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. Note that 3+4 and 4+3 count separately, because the two dice are different objects even when they look the same.

The “at least” column is useful in games where you need to beat a number: it adds up all the ways to get that total or higher.

Some useful facts that follow from the table:

• 7 is the single most likely total (1 in 6). This is why the number 7 plays a special role in many two-dice games: in Catan, for example, a 7 activates the robber instead of producing resources.
• 6 and 8 are the next most likely totals, each 5 in 36 (13.89%).
• Doubles (both dice the same) happen in 6 of the 36 outcomes, so 1 in 6, exactly as often as a total of 7.
• The average total is 7, the sum of two d6 averages (3.5 + 3.5).

Three six-sided dice (3d6)

With three dice there are 216 outcomes and the totals range from 3 to 18. The middle totals 10 and 11 are the most likely, and the extremes are rare: you need all three dice to show 6 to get 18, which happens once in 216 rolls. Many classic role-playing games used 3d6 to generate character abilities.

The more dice you add, the more the results bunch up around the average. That is why rolling several dice feels more predictable than rolling one big die with the same range: 3d6 and 1d16+2 both give results from 3 to 18, but with 1d16+2 an 18 is as likely as a 10.

The chance of rolling at least one six

Many games ask whether at least one die shows a particular number. Counting those outcomes directly is awkward, so use the opposite event instead. The chance that a single die is not a six is 5/6. The chance that none of n dice is a six is (5/6)ⁿ, so:

Chance of at least one six with n dice = 1 − (5/6)ⁿ

A common mistake is to think that six dice guarantee a six because each die has a 1 in 6 chance. In fact about one roll in three with six dice has no six at all.

The d20, advantage and disadvantage

In Dungeons & Dragons fifth edition you usually roll a d20 and try to reach a target number. Each face has a 5% chance, so the chance of rolling at least T is (21 − T) ÷ 20. With advantage you roll two d20 and keep the higher (2d20kh1); with disadvantage you keep the lower (2d20kl1).

If p is the normal chance of success, advantage succeeds unless both dice fail, so its chance is 1 − (1 − p)². Disadvantage needs both dice to succeed, so its chance is p².

Advantage helps most for middle targets: needing an 11, it raises your chance from 50% to 75%. The average of a d20 rolled with advantage is 13.825, and with disadvantage it is 7.175, compared with 10.5 for a normal roll.

Finding the average of any roll

The average (also called the expected value) of a roll is easy to work out, even for complicated notation, because averages simply add up:

Average of NdS + M = N × (S + 1) ÷ 2 + M

• 2d6+3: 2 × 3.5 + 3 = 10
• 8d6: 8 × 3.5 = 28
• 1d8+2d6+4: 4.5 + 7 + 4 = 15.5
• 1d12 averages 6.5 and 2d6 averages 7. The ranges are similar, but 2d6 is far more consistent and can never roll a 1.

“Keep” rolls do not follow the simple formula, because dropping a die changes the result in a way that depends on the other dice. For those, count every outcome. For example, checking all 1,296 outcomes of 4d6kh3 (roll four dice, keep the highest three) gives an average of about 12.24, a little higher than the 10.5 average of a plain 3d6.

Odds in well-known dice games

• Five of a kind with five dice in one throw (a “Yatzy” or “Yahtzee”): there are 6 ways out of 7,776, so the chance is 1 in 1,296, about 0.08%. In the real games you get up to three throws and can keep dice between throws, which makes it much more likely over a whole game.
• Doubles with two dice, which give an extra turn in Monopoly: 1 in 6 (16.67%).
• A natural 20 on a d20: 1 in 20 (5%) on a normal roll, 39 in 400 (9.75%) with advantage.

Dice have no memory

Each roll of a fair die is independent of every roll before it. After five rolls without a six, the chance of a six on the next roll is still exactly 1 in 6. Believing that a number is “due” after a long absence is a well-known mistake called the gambler’s fallacy. Over thousands of rolls the results do even out in proportion, but not because the dice try to catch up: the early streak simply becomes a smaller and smaller part of the whole.

Want to see this for yourself? Roll 100d6 on the dice roller a few times and see how close each total stays to the average of 350, then compare that with how much single rolls of 1d6 jump around. More questions are answered in the FAQ.