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Dice Probability Calculator
Choose how many dice you roll and how many sides they have, then pick a target: exactly, at least, at most or between two sums. You get the exact probability, the number of winning outcomes and a chart of every possible total.
Last updated:
d4, d6, d8, d10, d12, d20 or any size up to 100
Enter your values to see the result.
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How to use the Dice Probability Calculator
- Enter the number of dice and the number of sides (6 for normal dice, 20 for a d20).
- Choose a condition: exactly, at least, at most, between or not equal.
- Enter the target sum (and the upper sum for a range).
- Read the probability, the winning outcomes and the odds.
- Check the chart and table for every possible total.
Dice probability formulas
- Total outcomes for n dice with s sides: sⁿ (two d6: 36, three d6: 216).
- P(sum = k) = (number of ways to make k) ÷ sⁿ. The ways are the coefficients of (x + x² + … + xˢ)ⁿ.
- P(sum ≥ k) = P(k) + P(k + 1) + … + P(n·s), and P(sum ≤ k) = 1 − P(sum ≥ k + 1).
- Mean = n(s + 1)/2, variance = n(s² − 1)/12.
Two six-sided dice (2d6)
| Sum | Ways | Probability | At least this sum |
|---|---|---|---|
| 2 | 1 | 2.78% | 100% |
| 3 | 2 | 5.56% | 97.22% |
| 4 | 3 | 8.33% | 91.67% |
| 5 | 4 | 11.11% | 83.33% |
| 6 | 5 | 13.89% | 72.22% |
| 7 | 6 | 16.67% | 58.33% |
| 8 | 5 | 13.89% | 41.67% |
| 9 | 4 | 11.11% | 27.78% |
| 10 | 3 | 8.33% | 16.67% |
| 11 | 2 | 5.56% | 8.33% |
| 12 | 1 | 2.78% | 2.78% |
With 3d6 the most likely sums are 10 and 11 (27 ways each out of 216, 12.5%), and 18 has a chance of only 1/216 ≈ 0.46%.
Frequently asked questions
What is the probability of rolling a 7 with two dice?
6 of the 36 outcomes add up to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so the probability is 6/36 = 1/6 ≈ 16.67%. Seven is the most likely sum with two six-sided dice.
How do you calculate dice sum probabilities?
Count the ways to reach each sum and divide by the total number of outcomes, sidesⁿ. The counts come from convolving the distribution of one die with itself n times, which is the same as expanding (x + x² + … + xˢ)ⁿ. This tool does that exactly in your browser.
What does 'at least' mean for dice?
'At least 10' means a sum of 10 or more. Its probability is the sum of P(10), P(11) and so on up to the maximum. 'At most' is the opposite direction, and P(at least k) = 1 − P(at most k − 1).
What is the average roll of several dice?
The expected sum of n dice with s sides is n × (s + 1) ÷ 2, for example 3.5 for one d6, 7 for 2d6 and 10.5 for 3d6. The standard deviation is √(n × (s² − 1) ÷ 12).
Why do more dice make middle sums more likely?
There are many more combinations that add up to middle values than to extreme ones. As you add dice, the distribution becomes bell-shaped and approaches a normal distribution (the central limit theorem).
What is the chance of rolling 15 or higher on a d20?
Six faces (15–20) out of 20 qualify, so the probability is 6/20 = 30%.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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