Dice Probability Calculator
Die type:
Full Result
Distribution
Probability Distribution
For 2d6, totals range from 2 to 12. The most likely total is 7 at 16.67%; totals 2 and 12 are least likely at 2.78% each.
Show full distribution table
| Total | Ways | Probability | Cumulative P |
|---|
Calculate the probability that a particular face value appears a certain number of times when rolling multiple dice. This uses the binomial distribution, since each die independently either shows the target face or does not.
Face Event Result
How Dice Probability Works
When you roll one fair die, every face is equally likely. Roll a d6 and each face from 1 to 6 has exactly a 1/6 chance. This is basic probability: count the favorable outcomes, divide by the total. That logic holds no matter how many sides the die has.
When you add more dice, the total number of outcomes grows fast. Two d6 have 6 × 6 = 36 ordered outcomes, not 12. Three d6 have 6³ = 216. For n fair s-sided dice, the total is always sn. The word "ordered" matters: the roll (1, 6) counts separately from (6, 1), even though both produce the same sum of 7.
Why Sums Are Not Equally Likely
This is the most common mistake in dice probability. Two d6 can produce sums from 2 to 12 — that is 11 possible totals. But they do not each occur with probability 1/11. Rolling a 7 can happen six different ways; rolling a 2 can happen only one way. The underlying (ordered) outcomes are equally likely; the sums are not, because different sums have different numbers of ordered combinations behind them.
| Sum | Ways (ordered) | Probability | Combinations |
|---|---|---|---|
| 2 | 1 | 1/36 = 2.78% | (1,1) |
| 3 | 2 | 2/36 = 5.56% | (1,2) (2,1) |
| 4 | 3 | 3/36 = 8.33% | (1,3) (2,2) (3,1) |
| 5 | 4 | 4/36 = 11.11% | (1,4) (2,3) (3,2) (4,1) |
| 6 | 5 | 5/36 = 13.89% | (1,5) (2,4) (3,3) (4,2) (5,1) |
| 7 | 6 | 6/36 = 16.67% | (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) |
| 8 | 5 | 5/36 = 13.89% | (2,6) (3,5) (4,4) (5,3) (6,2) |
| 9 | 4 | 4/36 = 11.11% | (3,6) (4,5) (5,4) (6,3) |
| 10 | 3 | 3/36 = 8.33% | (4,6) (5,5) (6,4) |
| 11 | 2 | 2/36 = 5.56% | (5,6) (6,5) |
| 12 | 1 | 1/36 = 2.78% | (6,6) |
The distribution is symmetric around 7, which is also the expected value for 2d6. This symmetry holds for any number of identical fair dice.
Cumulative Probabilities: At Least and At Most
For at least a total, sum the probabilities from the target through the maximum. For at most, sum from the minimum through the target. The two are related through the complement rule: P(sum ≥ t) = 1 − P(sum ≤ t−1). These identities are useful for checking your work.
At Least One Specific Face
To find the probability that a particular face appears on at least one die, the complement rule is the cleanest route. P(at least one) = 1 − P(none). For n dice each with s sides, P(none show face f) = ((s−1)/s)n.
A common error is to multiply the one-die probability by the number of dice: 1/6 × 3 = 1/2. That is wrong because it counts overlap between dice (each die already having a 6 appears twice). The complement approach avoids this entirely. For more on counting methods, see permutations and combinations.
Exactly k Dice Showing a Face
When the question is "exactly k dice show a specific face," the situation is a binomial distribution problem. Each die is an independent trial with success probability 1/s. The formula is:
Expected Total and Variance
For n fair s-sided dice, the expected (average) total is n(s+1)/2. For a single d6 this is 3.5, and for 2d6 it is 7 — matching the peak of the distribution. The expected value calculation is independent of whether you are looking at sums or face counts.
Expected total
E[S] = n(s+1)/2
Variance of sum
Var(S) = n(s²−1)/12
Worked Examples
1. Single d6: Rolling exactly a 4
Favorable outcomes: 1 (just the face showing 4). Total outcomes: 6. P = 1/6 ≈ 16.67%.
2. Single d20: Rolling at least 15
Valid faces: 15, 16, 17, 18, 19, 20 — that is 6 outcomes. Total: 20. P = 6/20 = 3/10 = 30%.
3. 2d6: Sum exactly equals 7
Six ordered pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) give the sum 7 out of 36 total outcomes. P = 6/36 = 1/6 ≈ 16.67%. This is why 7 is the most common single total on 2d6.
4. 2d6: Sum between 5 and 8 inclusive
Ways for sums 5, 6, 7, 8: 4 + 5 + 6 + 5 = 20 out of 36. P = 20/36 = 5/9 ≈ 55.56%.
5. 3d6: At least one 6
P = 1 − (5/6)³ = 1 − 125/216 = 91/216 ≈ 42.13%.
6. 4d6: Exactly two dice show a 6
C(4,2)(1/6)²(5/6)² = 6 × 1/36 × 25/36 = 150/1296 = 25/216 ≈ 11.57%.
Common Mistakes
| Mistake | Why it's wrong | Correct approach |
|---|---|---|
| Assuming 11 sums on 2d6 are equally likely | Sums have different numbers of ordered combinations behind them | Count the ordered ways for each sum and divide by sn |
| Thinking 2d6 has only 12 outcomes (6 + 6) | The correct count is 6 × 6 = 36 ordered outcomes | Multiply sides for independent dice: total = sn |
| Multiplying 1/6 by 3 for "at least one 6 on 3d6" | This double-counts scenarios where multiple dice show a 6 | Use the complement: 1 − (5/6)³ |
| Confusing a sum event with a face event | "Total = 6" is not the same as "one die shows 6" | Use Sum Events for totals; Face Events for individual die faces |
| Treating impossible events as input errors | P(2d6 = 20) = 0 is a valid, correct answer, not an error | Return probability 0 with an explanation |
Frequently Asked Questions
For fair dice, divide the number of favorable ordered outcomes by the total number of ordered outcomes. For n dice with s sides, total outcomes = sn. Count how many ordered combinations produce your target total (using the convolution method for larger numbers), then divide. This tool does that counting exactly without simulation.
Six distinct ordered pairs add up to 7 on 2d6: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). No other sum on 2d6 has more combinations. The sum distribution is symmetric around the expected value of 7, which is n(s+1)/2 = 2 × 7/2 = 7.
No. Individual face outcomes are equally likely, but sums generally are not. On 2d6, the sum 7 occurs with probability 1/6, while the sum 2 occurs with probability 1/36. Different totals have different numbers of ordered outcome combinations, which is why the distribution peaks at the middle and tapers toward the extremes.
It depends on how many dice you roll. The fastest method is the complement: P(at least one 6) = 1 − P(no 6 on any die) = 1 − (5/6)n. For n = 1: 16.67%. For n = 2: 30.56%. For n = 3: 42.13%. For n = 6: 66.51%.
Exact math. The sum distribution is computed using dynamic programming on exact integer counts, and the result is expressed as a reduced fraction. No random sampling is involved. This means the answer is always the true theoretical probability, not an estimate that varies between runs.
This calculator answers: "What fraction of all possible rolls gives this result?" The Dice Roller simulates an actual roll and gives you one random outcome. Use the probability calculator when you want to know the theoretical odds; use the dice roller when you want to generate a random result for a game.
Yes. Click the die-type buttons (d4, d6, d8, d10, d12, d20, d100) to set the number of sides, or type any value from 2 to 100 in the Sides per Die field. The calculator handles any combination of dice count and sides.
Two six-sided dice have 6 × 6 = 36 ordered outcomes. Each die independently lands on 1 through 6, and (1,6) counts as a different outcome from (6,1). This is why the total is 36, not 6 + 6 = 12 or 11 possible sums. For n dice each with s sides, total ordered outcomes = sn.
For n fair s-sided dice, the expected total is n(s+1)/2. For a single d6 this is (6+1)/2 = 3.5. For 2d6 it is 2 × 3.5 = 7, which is why 7 is the peak of the 2d6 distribution. For 1d20 it is (20+1)/2 = 10.5. The Dice Average Calculator covers expected values in more detail.
"Total = 6" asks how often all dice together add up to 6 — this is a sum event. "At least one die shows a 6" asks how often a specific face appears on any die — this is a face event. They use different formulas and are calculated separately in this tool. Confusing the two is one of the most common errors in dice probability.