About the Dice Probability Calculator
This dice probability calculator works out the chance of rolling a particular total with any number of dice. Choose how many dice you roll, how many sides each has, the target sum and whether you need exactly that total, at least it, or at most it. You get the probability as a percentage, the “1 in N” odds, the number of winning combinations out of all possible outcomes, and a chart of the full distribution of sums.
It is built for board gamers (Catan, Monopoly, craps), tabletop role-players checking a d20 skill check or a 3d6 or 8d6 damage roll, game designers balancing mechanics, and students learning probability. The distribution chart shows why totals near the middle are so much more common than extremes when you add several dice together.
The math assumes fair dice numbered 1 to the number of sides, rolled independently, and adds their faces. Up to 30 dice with up to 100 sides each are supported. Results are exact (computed by convolution, not simulation): the numbers of winning combinations and total outcomes are exact whole numbers even when they run to dozens of digits, as with 30d100 (10^60 outcomes).
With the default inputs, the probability is 16.6667%. Change any value above to recalculate instantly.
How to use the dice probability calculator
- 1Enter how many dice you are rolling.
- 2Enter the sides per die (6 for standard dice, 20 for a d20).
- 3Choose exactly, at least, at most, more than or less than.
- 4Enter the target total.
- 5Read the probability and odds, and use the chart to compare every possible total.
Formula and method
Each of the sⁿ ordered outcomes of n fair s-sided dice is equally likely, so a probability is the number of outcomes that give the wanted total divided by sⁿ. The count of ways to reach a total t can be written with the inclusion–exclusion formula above, but it is computed here by convolution in exact integer arithmetic: the distribution for one die is uniform, and each extra die spreads every existing total evenly across the next s totals.
“At least”, “at most” and similar conditions simply add the probabilities of all qualifying totals. The average total is n(s + 1)/2 and, for two or more dice, the most likely totals sit at that average.
- n
- Number of dice
- s
- Sides per die (faces numbered 1 to s)
- t
- Target total
- C(n, k)
- Binomial coefficient “n choose k”
Worked examples
Rolling a 7 with two six-sided dice
Six of the 36 combinations add to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so the probability is 6/36 = 16.67%, or 1 in 6. Seven is the most common total with 2d6.
At least 15 on 3d6
Totals of 15, 16, 17 and 18 can be made in 10, 6, 3 and 1 ways — 20 of 216 outcomes, a 9.26% chance (1 in 10.8).
A d20 roll of 15 or higher
Six faces (15–20) of a 20-sided die succeed, so the chance is 6/20 = 30%.
At most 10 on 4d6
Out of 1,296 outcomes for four dice, 206 total 10 or less, a probability of about 15.90%. The average of 4d6 is 14, so low totals like this are fairly uncommon.
At least 100 on 25d6 (exact big counts)
Twenty-five dice have 6^25 ≈ 2.84 × 10^19 outcomes — too many for ordinary floating point to count exactly — so the counts are shown as exact whole numbers: 2,283,244,349,009,314,473 of them total 100 or more, an 8.03% chance (about 1 in 12.45). The average of 25d6 is 87.5.
Frequently asked questions
What is the most common roll with two dice?+
Seven. It can be made six ways out of 36 (1+6, 2+5, 3+4 and the reverses), a 16.67% chance. Totals of 2 and 12 are the rarest, with only one way each (2.78%).
How do you calculate dice probability?+
Count the outcomes that give the result you want and divide by the total number of outcomes, which is sides^dice. With 2d6 there are 6² = 36 outcomes; for several dice, counting totals by hand gets tedious, so a calculator or convolution is used.
What is the probability of rolling doubles with two dice?+
There are 6 doubles (1-1 through 6-6) out of 36 outcomes, so the probability is 6/36 = 1/6 ≈ 16.67%. Rolling a specific double, like double sixes, is 1/36 ≈ 2.78%.
What is the average roll of 3d6?+
Each die averages (1 + 6)/2 = 3.5, so three dice average 10.5. In general n dice with s sides average n(s + 1)/2 — for example 8d6 averages 28 and 2d20 averages 21.
Why do more dice make middle totals more likely?+
Middle totals can be built from many different combinations while extreme totals need every die to land high or low. As you add dice the distribution becomes bell-shaped, approaching a normal distribution (central limit theorem).