Dice probability rests on one idea: every face of a fair die is equally likely, and dice do not affect each other. A single six-sided die shows any given number with probability 1/6. For several dice you either multiply probabilities (all dice must do something), use the binomial formula (a certain number of dice must do something) or count the ways to make a total (sums).
Sums are where intuition goes wrong. Two dice can land in 36 equally likely ways, but only one of them gives a 2 while six give a 7, so 7 is six times as likely as 2. With more dice the counting is done by convolution: the distribution of the sum is built up one die at a time. That is exactly what this calculator does, for up to 100 dice with 2 to 1,000 sides.
Pick the dice, choose the question and enter the target. You get the probability as a percentage, the odds as “1 in X”, the number of favourable outcomes out of the total, the chance that it does not happen and a chart of the relevant distribution. For two dice you also see every one of the outcomes in a grid with the favourable ones marked.
How to use the Dice Probability Calculator
- Dice type
- Number of sides (custom die)
- Number of dice
- I want…
- Face value
- Target sum
- X, how many dice must show the value
- Roll type
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Each condition uses its exact formula: powers for “every die” conditions, the binomial distribution for “X dice show a value”, exact convolution of the dice distributions for sums, and the distribution of the higher or lower of two dice for advantage and disadvantage. Nothing is simulated, so the results are exact.
Worked example
Two six-sided dice, sum equal to 7. The favourable pairs are (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1): 6 of 36, so P = 1/6 ≈ 16.67%, or 1 in 6. For “sum at least 10” the pairs are (4,6), (5,5), (6,4), (5,6), (6,5) and (6,6): 6/36 again. For a d20 with advantage, the chance of 15 or more is 1 − (14/20)² = 1 − 0.49 = 51%.
Two-dice totals
| Total | Ways | Probability |
|---|---|---|
| 2 or 12 | 1 | 2.78% |
| 3 or 11 | 2 | 5.56% |
| 4 or 10 | 3 | 8.33% |
| 5 or 9 | 4 | 11.11% |
| 6 or 8 | 5 | 13.89% |
| 7 | 6 | 16.67% |
Formula
One face: P = 1/s · All n dice ≥ v: ((s − v + 1)/s)ⁿ · Exactly X dice show v: C(n, X)(1/s)ˣ(1 − 1/s)ⁿ⁻ˣ · P(sum = v) = (ways to make v) ÷ sⁿ · Advantage: P(max ≥ v) = 1 − ((v − 1)/s)²
Worked example
Inputs: Dice type: Cube – d6 (6 faces); Number of dice: 2; I want…: The sum is exactly…; Target sum: 7.
Results: Probability = 16.6667 %; What was calculated = Probability that the sum is exactly 7; Odds (1 in …) = 1 in 6; Favourable outcomes = 6 of 36 equally likely outcomes; Probability it does not happen = 83.3333 %; All 2-dice outcomes = 1st\2nd 1 2 3 4 5 6
────────────────────────────────────────
1 2 3 4 5 6 7*
2 3 4 5 6 7* 8
3 4 5 6 7* 8 9
4 5 6 7* 8 9 10
5 6 7* 8 9 10 11
6 7* 8 9 10 11 12
(* = outcome counts; each cell shows the sum of the two dice).
Key terms
- Favourable outcome
- An outcome that satisfies the condition you are asking about.
- Independent events
- Events where one result does not change the chance of the other, like separate dice.
- Convolution
- Combining the distributions of two dice to get the distribution of their sum.
- Advantage / disadvantage
- Rolling two d20s and keeping the higher (advantage) or lower (disadvantage) result, as in Dungeons & Dragons 5th edition.
Frequently asked questions
What is the probability of rolling a 7 with two dice?
6/36 = 1/6 ≈ 16.67%, the most likely total. 2 and 12 are the least likely at 1/36 ≈ 2.78% each.
What is the chance of rolling an 8 in Catan?
5/36 ≈ 13.89% on each roll of two dice, the same as rolling a 6.
What is the chance of a 6 on at least one of two dice?
1 − (5/6)² = 11/36 ≈ 30.56%.
What is the chance that three dice all show 4 or more?
(3/6)³ = 1/8 = 12.5%.
What is the chance of a Yahtzee in one roll?
Five dice all equal: 6/6⁵ = 1/1,296 ≈ 0.077%.
How does advantage change a d20 roll?
Rolling two d20s and keeping the higher makes 15 or more 51% likely instead of 30%; disadvantage cuts it to 9%.
What is the most likely total of three dice?
10 and 11, each with 27/216 = 12.5%.
How many outcomes do n dice have?
sⁿ: 36 for two d6, 216 for three, 1,296 for four.