Calculate two-dice probabilities for sums, doubles, products, and differences. View complete sum distribution, outcome grid, cumulative probabilities, and custom-sided dice support.
The two dice probability calculator computes the probability of any outcome when rolling two dice. Calculate the chance of specific sums, doubles, products, differences, or getting at least one particular face value. Supports standard 6-sided dice and custom dice with any number of sides.
For two standard d6 dice, there are 36 equally likely outcomes (6 × 6). The most common sum is 7 with 6 ways to roll it (16.67%), while 2 and 12 each have only 1 way (2.78%). This calculator shows the full distribution, outcome grid, cumulative probabilities, and what happens over multiple rolls.
Choose from preset queries like "Sum = 7", "Any doubles", or "At least one 5", or select from eight different calculation modes with custom target values. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
Two-dice probability is one of the most common probability scenarios — from board games and gambling to classroom exercises. This calculator handles every possible question about two dice: sums, doubles, products, differences, and more. Support for custom dice sizes makes it useful for tabletop RPG players and game designers.
The visual distribution, outcome grid, and cumulative tables make it an excellent teaching tool for understanding discrete probability distributions.
P(sum = k) = (number of ways to roll k) / (sides₁ × sides₂). For 2d6: ways to roll sum k = k − 1 for k ≤ 7, and 13 − k for k > 7. P(doubles) = min(sides₁,sides₂) / (sides₁ × sides₂).
Result: 16.67% (6/36), odds 1:5
Sum of 7 can be rolled 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Out of 36 total outcomes, that's 6/36 = 16.67%. The chance of rolling at least one 7 in 3 rolls is 42.1%.
The sum of two d6 dice follows a triangular distribution: probabilities increase from 2 to 7, then decrease from 7 to 12. This happens because there are more ways to make middle values. Only (1,1) makes 2, but six different pairs make 7. This is the simplest example of the central limit theorem — the sum of uniform random variables tends toward a bell shape.
A common misconception is that if you haven't rolled a 7 in several tries, you're "due" for one. Each roll is independent — the probability remains 16.67% regardless of history. However, over many rolls, the law of large numbers guarantees the proportion of 7s will converge to 16.67%.
Interestingly, it's possible to design three dice A, B, C where A beats B more than half the time, B beats C more than half the time, but C beats A more than half the time. These "non-transitive dice" demonstrate that probability comparisons aren't always transitive, a concept relevant to voting theory and tournament design.
For two d6 dice, 7 is most likely (6 ways out of 36 = 16.67%). For two dice with s₁ and s₂ sides, the most likely sum is around (s₁+s₂+2)/2.
For d6+d6: sums 2-7 have 1,2,3,4,5,6 ways respectively. Sums 8-12 have 5,4,3,2,1 ways (symmetric). The pattern forms a triangle: increasing to 7, then decreasing.
With two d6 dice, there are 6 possible doubles (1-1 through 6-6) out of 36 outcomes = 16.67% or 1/6. Each specific double has probability 1/36 ≈ 2.78%.
Yes, each die roll is completely independent of the other. The first die showing a 6 has no effect on what the second die shows. This is why P(double 6) = P(6)×P(6) = 1/36.
In Monopoly, you move the sum of two dice, so knowing sum probabilities helps predict which properties get landed on most. In Craps, the come-out roll probabilities (7 or 11 wins, 2/3/12 loses) directly determine house edge.
The calculator supports any combination. A d4+d8 has 32 outcomes with sums from 2 to 12. The distribution is no longer symmetric if the dice have different sizes. Custom dice are common in tabletop RPGs like D&D.