Dice Probability Calculator
Pick the dice and a target total, and this dice probability calculator gives the exact chance of rolling exactly, at least or at most that total, with every total's odds in a table.
Chance of exactly 7
- As a fraction Ways to make it over all outcomes, then in lowest terms.
- 6/36 = 1/6
- As 1 in N
- 1 in 6
- Ways it can happen 2 dice of 6 sides have 6^2 = 36 equally likely outcomes, each die counted separately.
- 6 of 36
- Odds against Outcomes that miss to outcomes that hit.
- 5 to 1
- Possible totals
- 2 to 12
- Mean total 2 x (6 + 1) / 2. Each d6 averages 3.5.
- 7
- Standard deviation The square root of 2 x (6^2 - 1) / 12.
- 2.42
- Most likely total
- 7
| Total | Ways | Probability | This or less | This or more |
|---|---|---|---|---|
| 2 | 1 | 2.78% | 2.78% | 100% |
| 3 | 2 | 5.56% | 8.33% | 97.22% |
| 4 | 3 | 8.33% | 16.67% | 91.67% |
| 5 | 4 | 11.11% | 27.78% | 83.33% |
| 6 | 5 | 13.89% | 41.67% | 72.22% |
| 7 | 6 | 16.67% | 58.33% | 58.33% |
| 8 | 5 | 13.89% | 72.22% | 41.67% |
| 9 | 4 | 11.11% | 83.33% | 27.78% |
| 10 | 3 | 8.33% | 91.67% | 16.67% |
| 11 | 2 | 5.56% | 97.22% | 8.33% |
| 12 | 1 | 2.78% | 100% | 2.78% |
How the Dice Probability Calculator works
Every dice roll probability is a count. Each die lands on one of its faces with equal chance, so n dice with s sides have s^n outcomes, all equally likely, and the chance of a total is the number of outcomes that make it divided by s^n. Two six-sided dice have 6 x 6 = 36 outcomes. Six of them add up to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so the chance of a 7 is 6/36 = 1/6, or 16.67%.
For 2 dice the probability chart is short enough to learn. Out of 36, the totals 2 through 12 come up 1, 2, 3, 4, 5, 6, 5, 4, 3, 2 and 1 ways. A 7 is the most likely roll, 2 and 12 the least likely at 1/36 each, and the pattern is symmetric around 7. Rolling 7 or more covers 21 of the 36 outcomes, which is 58.33%.
More dice pile up in the middle. Three dice have 216 outcomes, and 10 and 11 are tied as most likely with 27 ways each, 27/216 = 1/8 or 12.5%. Rolling at least 10 on three dice is 135/216 = 5/8, or 62.5%. An 18 takes three sixes: 1 in 216.
For any number of dice, this calculator counts by convolution: the ways to make each total with three dice are built from the counts for two dice, one die at a time. That is exact in whole numbers for any pool up to 20 dice of 1,000 sides, so the fraction shown is the true one and the percentage is rounded only for display. The table lists each total's ways and probability, plus the cumulative chance of that total or less and that total or more.
The mean total of n dice with s sides is n(s + 1)/2, so 2d6 averages 7 and 3d6 averages 10.5. The standard deviation is the square root of n(s^2 - 1)/12: 2.42 for 2d6 and 2.96 for 3d6.
Dice probabilities in Yahtzee follow the same counting. Five dice have 6^5 = 7,776 outcomes, and six of them are five of a kind, so a Yahtzee on one roll is 6/7,776, or 1 in 1,296. With three rolls, keeping the most common face each time and rerolling the rest, the chance rises to 347,897/7,558,272, which is 4.6%, about 1 in 21.7. That figure comes from a small exact Markov chain whose state is how many dice show the most common face.
Dice Probability Calculator FAQ
What is the most common roll with two dice?
7. It can be made 6 ways out of 36, so it comes up 1/6 of the time, 16.67%. 6 and 8 are next at 5/36 each, and 2 and 12 are the rarest at 1/36.
What are the odds of rolling a Yahtzee?
1 in 1,296 on a single roll of five dice: 6 of the 7,776 outcomes are five of a kind. Within the three rolls of a turn, keeping the most common face and rerolling the rest, it is 347,897/7,558,272, or 4.6%, about 1 in 21.7. Within two rolls it is 221/17,496, 1.26%.
What is the probability of rolling a 10 with three dice?
27/216 = 1/8, or 12.5%. 10 and 11 are the most likely totals on three six-sided dice, with 27 of the 216 outcomes each. The chance of 10 or more is 135/216, 62.5%.
What is the difference between exactly, at least and at most?
Exactly counts only the target total. At least counts the target and everything above it, which is what a roll against a target number needs. At most counts the target and everything below. Exactly 7 on 2d6 is 6/36; at least 7 is 21/36; at most 7 is also 21/36.
Why are some totals more likely than others?
More combinations of faces add up to them. One die is flat, every face 1 in s. With two or more, a middle total can be made many ways and an extreme total only one: on 2d6, a 12 needs two sixes, while a 7 has six different ways.