Article — Dice Probability Calculator
Dice Probability Calculator
Dice probability is the chance of a specific outcome when rolling one or more fair dice. For two six-sided dice, the chance of rolling a seven is 6/36 = 16.67%, the most likely sum. The exact formula for n s-sided dice is P(S = t) = c_t / s^n, where c_t counts the ordered tuples summing to t.
This page covers the math behind every dice roll: how to count outcomes for any number of dice, why the sum distribution bends into a bell curve, what to do when you need an at-least or at-most probability, and where common intuition fails. Every value is computed by exact enumeration in the calculator, not a Monte Carlo approximation.
What is dice probability?
Dice probability is the ratio of favorable outcomes to total outcomes when rolling fair dice. A fair die has equal probability on each face. For a standard six-sided die, every face has chance 1/6 ≈ 16.67%. For a twenty-sided die used in Dungeons & Dragons, each face has chance 1/20 = 5%.
Two ideas anchor the rest: the sample space (every outcome that can happen) and the event (the subset you care about). For two d6, the sample space has 36 ordered pairs. If you care about pairs summing to 7, the event has 6 elements. The probability is 6/36.
Galileo wrote one of the first known dice probability essays around 1620, settling a gambling dispute. He showed that rolling a 10 with three dice is slightly more likely than rolling a 9, even though both can be made in six unordered ways. The ordered count, which is what matters, is 27 ways for 10 versus 25 ways for 9 out of 216 total.
Dice probability for one die
For one fair s-sided die, P(X = k) = 1/s for any k in {1,..., s}. That uniform shape is the building block for every multi-die calculation. The expected value is (s + 1) / 2: 2.5 for a d4, 3.5 for a d6, 10.5 for a d20.
Variance for a single die equals (s^2 − 1) / 12. For a d6 that is 35/12 ≈ 2.917; the standard deviation is the square root, about 1.708. A d20 has variance 33.25 and SD about 5.77, which is why a single d20 roll feels swingier than a single d6.
Dice probability for the sum of multiple dice
For n dice the total number of outcomes is s^n. For 2d6, that is 36. For 3d6, 216. For 4d6, 1,296. To get the probability of a sum, count the ordered tuples that hit the target and divide by s^n. With 2d6 the count by sum is the familiar triangle: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 for sums 2 through 12.
For more than two dice, the count is harder to do by hand. The calculator builds the full distribution by convolution: multiply the single-die generating polynomial (x + x^2 +... + x^s) by itself n times and read off the coefficients. That gives the exact count for every sum in a few microseconds, even for 10d20.
2 → 1 3 → 24 → 3 5 → 46 → 5 7 → 68 → 5 9 → 410 → 3 11 → 212 → 1 total 36Expected value and variance of dice rolls
Linearity of expectation makes the mean of a sum easy: E[S] = n * E[X] = n * (s + 1) / 2. Two d6 average to 7. Three d20 average to 31.5. For 4d6 keep highest three (a common D&D ability score roll) the mean is about 12.24, which is why the standard scores tend to cluster between 11 and 14.
Variance also adds for independent dice: Var(S) = n * (s^2 − 1) / 12. The standard deviation is the square root and grows like √n, not like n. That is the source of the bell shape: relative to the mean, multi-die sums concentrate more tightly as n increases.
Dice probability in tabletop RPGs
In D&D 5th edition, an ability check rolls 1d20 + modifier against a difficulty class. The chance of meeting or exceeding DC X with modifier M is P(d20 ≥ X − M). For modifier +5 against DC 15, that is P(d20 ≥ 10) = 11/20 = 55%. Advantage rolls 2d20 and keeps the higher; the success probability becomes 1 − ((X − M − 1)/20)^2 ≈ 79.75% for the same example.
For damage, the calculator handles the at-least and at-most modes you need. A fireball does 8d6 damage; the expected value is 28 and the SD is about 4.83. The chance of rolling 35 or higher on 8d6 is about 7.65% — a useful number for game balance and encounter design.
For the chance of "at least one success" across n independent trials, the cleanest formula is the complement: 1 − (1 − p)^n. Rolling a 20 on a single d20 has p = 0.05; the chance of at least one 20 across 10 attempts is 1 − 0.95^10 ≈ 40.13%.
Common dice probability mistakes
The most common error is the gambler's fallacy: assuming that because a 6 has not come up recently, it is "due." Dice have no memory. Each roll is independent. After ten non-sixes, the next roll has the same 1/6 chance as the first.
Another trap is counting unordered combinations instead of ordered ones. For 2d6 summing to 3, there are two ordered pairs (1,2) and (2,1) — not one. The sample space treats the dice as distinguishable, so always count ordered tuples. Galileo's three-dice paradox is exactly this trap.
The dice probability formulas on this page assume each die is independent. If you are drawing dice from a finite set without replacement (rare, but it happens in some board games), the calculations change. For ordinary dice rolling, independence always holds.
Monte Carlo simulation vs. exact enumeration
Many online dice calculators use Monte Carlo: roll the dice 100,000 times in code and count successes. That works, but it gives an approximation with sampling error roughly 1/√N. For 100k trials, error is about 0.3%. The calculator on this page uses exact enumeration via polynomial convolution, so the answer is the exact rational probability with no sampling noise.
A dice probability cheat sheet
- 1d6 specific face = 1/6 ≈ 16.67%
- 1d20 natural 20 = 1/20 = 5%
- 2d6 sum of 7 = 6/36 ≈ 16.67% (most likely sum)
- 2d6 doubles = 6/36 ≈ 16.67%
- 3d6 sum of 10 or 11 = 27/216 each ≈ 12.50%
- 4d6 drop lowest mean ≈ 12.24
- At least one 20 in n rolls = 1 − (19/20)^n
- Variance of nd6 = 35n/12, SD = √(35n/12)