Dice Probability Calculator
Calculate exact dice probabilities across nine different scenarios, from a specific sum to advantage rolls. Choose any number of dice, any number of sides (d4 through d100 and beyond), pick your probability mode, and get the result with a full show-your-work breakdown. The expected average, favorable outcome count, and plain-English interpretation are always shown.
Formula
Worked example
Two six-sided dice (n = 2, S = 6), sumEquals mode, target 7: there are 6 ways to make 7 (1+6, 2+5, 3+4 and their reverses) out of 6^2 = 36 rolls, so P = 6/36 = 16.667%. Advantage d20 >= 15: P(one die < 15) = 14/20 = 0.70, P(both fail) = 0.49, so P(advantage hits) = 51%. Expected sum for 2d6 = 2*(6+1)/2 = 7.
How dice probability works across all nine modes
Every die face is equally likely so rolling n dice with S sides produces exactly S to the power of n equally likely ordered outcomes. The simplest mode, "sum equals target," counts the ordered combinations that add up to that total using an inclusion-exclusion formula, then divides by the total outcome count. "Sum at least" and "sum at most" extend that by summing over all totals above or below the threshold. The "exactly K" and "at least K" modes use the binomial distribution: each die independently has a 1-in-S chance of showing a specific face, so the number of dice matching follows a binomial model with parameters n and 1/S. "All same" is a special case where any matching face counts, giving sides times (1/S)^n. The "all-high" and "all-low" modes raise the per-die probability (faces at or above, or at or below, the threshold divided by sides) to the power of the dice count.
Advantage and disadvantage (D&D and other RPGs)
Advantage means rolling two dice and keeping the higher result; disadvantage means keeping the lower. The probability of the higher die being at least a target value t on a d-sided die is 1 minus the probability both dice fall below t, which equals 1 minus ((t minus 1) divided by sides) squared. Disadvantage flips this: both dice must be at or above the target, so the probability is ((sides minus t plus 1) divided by sides) squared. These are D&D staples and are not available on most basic dice calculators. The shift in probability is dramatic: on a d20, the chance of rolling at least 15 is 30% normally, 51% with advantage, and only 9% with disadvantage.
Roll modifiers
Many RPG systems add a flat bonus or penalty to the dice total. The modifier input shifts the effective sum without changing the shape of the distribution. For example, rolling 1d6 with a +3 modifier has a range of 4 to 9, not 1 to 6. The calculator accounts for modifiers in the sum-based modes by subtracting the modifier from the target before applying the probability formula, which is mathematically identical to shifting the distribution. The expected average also updates to include the modifier, so your result cards always show the true final number.
Reading the favorable outcomes and odds
The "1 in N" odds figure is simply the total possible rolls divided by the favorable outcomes. For rare events this is more intuitive than a tiny percentage: "1 in 1296" is clearer than "0.077%" when rolling five sixes on 5d6. Because every ordered roll is weighted equally, the same method scales from a d4 to a d100 and from one die to twenty, the arithmetic never changes, only the size of the numbers. For advantage and disadvantage modes the total shown is sides squared (since two dice are always compared), and the favorable count is the number of two-die combinations where the relevant extreme meets the target.
Probability of each sum with two six-sided dice (2d6)
| Sum | Ways | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
The classic 2d6 distribution: 7 is the most likely total at 16.67%, the extremes are the least likely.
Frequently asked questions
What is the difference between "sum equals" and "sum at least" mode?
"Sum equals" gives the probability of hitting one specific total, for example exactly 7 with 2d6 is 16.67%. "Sum at least" adds up the probabilities of all totals at or above your threshold, for example 7 or higher with 2d6 is about 58.33%. Use "sum equals" when you need an exact number (a critical chart result) and "sum at least" when anything at or above a threshold counts as a success (an attack roll in a tabletop RPG).
How does advantage work in D&D, and how does this calculator handle it?
Advantage means rolling two dice and taking the highest result. The probability of the better die reaching a target value t on a d20 is 1 minus (t minus 1 divided by 20) squared. A target of 15 is 30% on a straight roll but 51% with advantage. The disadvantage mode is the same concept in reverse: both dice must hit the target, so the probability is ((20 minus t plus 1) divided by 20) squared, giving only 9% for the same target of 15. This calculator shows both modes for any die type, not just the d20.
Why is 7 the most common roll with two six-sided dice?
A total of 7 can be made six different ways (1+6, 2+5, 3+4 and their reverses), more than any other sum. Totals like 2 or 12 can only be made one way each, making them six times rarer. The probability of 7 is 6 divided by 36, which is exactly 16.67%. This pattern, a peak at the center and tapering toward the extremes, holds for any number of same-sided dice and is why the distribution approaches a bell curve as you add more dice.
What does a roll modifier do to the probability?
A flat modifier shifts the entire distribution up or down without changing its shape. Rolling 2d6 + 3 has the same probabilities as rolling 2d6, just with every total increased by 3, so the range becomes 5 to 15 and the most likely result is 10. The calculator accounts for modifiers in sum-based modes by adjusting the effective target before computing, and the expected average output already includes the modifier.
How do I find the chance that all dice roll a high value?
Use the "All dice show >= threshold" mode (all-high). Set the threshold to the minimum face value you want every die to show. For example, rolling 3d6 and needing each die to show 5 or 6: the per-die probability is 2/6 = 33.33%, and for all three dice it is (2/6)^3 = 3.70%. This is useful for all-or-nothing checks where every single die must succeed.