Two dice are the most common random device in board games, and their outcomes are easy to list: with six-sided dice there are 6 × 6 = 36 equally likely ordered pairs. Every two-dice probability is a count of favourable pairs divided by 36. Doubles are the six pairs on the diagonal (1/6), a sum of 7 is the six pairs on the opposite diagonal (also 1/6), and both dice at least 3 is a 4 × 4 block of 16 pairs (16/36 ≈ 44.44%).
The calculator does that counting for you for any condition and any die type – two d4s, two d10s, two d20s or a custom pair. It shows the probability, the odds, the number of favourable outcomes out of the total and the chance that it does not happen, and draws the matching chart. Most usefully, for dice with up to 12 sides it prints the complete grid of outcomes with each sum written in and the favourable cells starred, so you can check the count by eye.
For D&D players there is also advantage and disadvantage: two d20s with the higher or lower one kept.
How to use the Two Dice Probability Calculator
- Dice type
- Number of sides (custom die)
- I want…
- Face value
- Target sum
- X – how many dice must show the value
- Roll type
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
For two dice the calculator multiplies single-die probabilities for “both” conditions, uses the binomial formula for “exactly / at least X dice”, and counts sums exactly. The grid lists every ordered pair (first die down, second die across).
Worked example
Both dice at least 3: each die has 4 suitable faces (3, 4, 5, 6), so 4 × 4 = 16 of the 36 pairs qualify and P = 16/36 = 4/9 ≈ 44.44%. At least one die showing 6: 36 − 25 = 11 pairs, P ≈ 30.56%. Sum of 8: the pairs (2,6), (3,5), (4,4), (5,3), (6,2), P = 5/36 ≈ 13.89%.
Common two-dice probabilities
| Event | Favourable pairs | Probability |
|---|---|---|
| Doubles | 6 | 16.67% |
| Sum of 7 | 6 | 16.67% |
| Sum of 2 or 12 | 1 each | 2.78% |
| At least one 6 | 11 | 30.56% |
| Both dice ≥ 4 | 9 | 25% |
| Sum ≥ 8 | 15 | 41.67% |
Formula
Two dice with s sides: s² equally likely outcomes · P(both ≥ v) = ((s − v + 1)/s)² · P(at least one shows v) = 1 − ((s − 1)/s)² · P(sum = t) = ways ÷ s²
Worked example
Inputs — Dice type: Cube – d6 (6 faces); I want…: Every die shows at least…; Face value: 3.
Results — Probability = 44.4444 %; What was calculated = Probability that every die shows 3 or more; Odds (1 in …) = 1 in 2.25; Favourable outcomes = 16 of 36 equally likely outcomes; Probability it does not happen = 55.5556 %; All 2-dice outcomes = 1st\2nd 1 2 3 4 5 6
────────────────────────────────────────
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6* 7* 8* 9*
4 5 6 7* 8* 9* 10*
5 6 7 8* 9* 10* 11*
6 7 8 9* 10* 11* 12*
(* = outcome counts; each cell shows the sum of the two dice).
Key terms
- Ordered pair
- An outcome written as (first die, second die), so (2,5) and (5,2) are different.
- Outcome grid
- A table of all s × s ordered pairs used to count favourable outcomes.
- Independent events
- The result of one die does not affect the other, so probabilities multiply.
Frequently asked questions
What is the chance of rolling doubles with two dice?
6/36 = 1/6 ≈ 16.67%.
What is the chance of rolling two sixes?
1/6 × 1/6 = 1/36 ≈ 2.78%.
What is the chance that both dice show 3 or more?
(4/6)² = 16/36 ≈ 44.44%.
What is the chance of at least one 6 on two dice?
1 − (5/6)² = 11/36 ≈ 30.56%.
What is the chance of a sum of at least 10?
6/36 = 1/6 ≈ 16.67% (sums 10, 11 and 12 have 3 + 2 + 1 ways).
Why are there 36 outcomes and not 21?
Ordered pairs (first die, second die) are equally likely; 3-4 and 4-3 are different outcomes even if the dice look the same.
What is the chance of a sum of 8 or more?
15/36 ≈ 41.67%.