Calculates the probability of rolling a specific total with any number of same-sided dice — a classic combinatorics problem.
How it works
Every possible combination of dice outcomes is enumerated to count how many ways produce the target sum, then divided by the total number of possible outcomes.
What this does not include
This does not include the average (expected value) of a dice roll — for that, use this site’s dice average calculator instead.
How to use this calculator
- Enter the number of dice, sides per die, and your target sum.
A worked example
Two standard six-sided dice, target sum 7.
Each die has 6 faces, so there are 6 × 6 = 36 equally likely outcomes. Six of them total 7 — (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1) — so the probability is 6 ÷ 36 = 16.67%, or 1 in 6.
Compare that with a target of 10: only (4,6), (5,5) and (6,4) work, giving 3 ÷ 36 = 8.33% — exactly half as likely. The ordered pairs matter: (4,6) and (6,4) are two distinct outcomes, which is why 7 beats 10 despite both looking like “a few combinations.”
What the variables mean
| Variable | Meaning | In the example |
|---|---|---|
| Number of dice | How many dice are rolled together | 2 |
| Sides per die | Faces on each die; all dice are assumed identical | 6 |
| Target sum | The total you want the dice to add up to | 7 |
| Total outcomes | Sides raised to the power of the dice count | 6² = 36 |
Edge cases worth knowing
Unreachable sums return zero, not an error. Two six-sided dice cannot total 1 or 13 — the possible range is 2 to 12 — so those targets have a genuine probability of 0%.
The distribution is symmetric, and peaks in the middle. With any number of fair dice, the most likely total sits at the centre of the possible range, and totals equidistant from that centre are equally likely: with two dice, 6 and 8 share the same probability, as do 5 and 9.
More dice means a sharper peak. Adding dice makes extreme totals rapidly less likely relative to central ones, because there are far more ways to arrange a middling sum than an extreme one.
Frequently asked questions
Why is rolling a 7 with two six-sided dice more likely than rolling a 2?
Because there are six different combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) that add up to 7, but only one combination (1+1) that adds up to 2.
What’s the most likely sum with two six-sided dice?
7 — it sits in the middle of the possible range (2 to 12) and has the most combinations that produce it.
Does this work for non-standard dice, like 20-sided dice?
Yes — enter any number of sides per die, and the calculator enumerates the combinations accordingly.
