Dice Probability Calculator Guide
Calculate probabilities for any dice combination — sum distributions, advantage rolls, and expected values.
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Tags: dice probability calculator, dice roll probability guide, D&D probability math
Dice Probability Calculator Guide Part of our complete guide to this topic — see the full series. Dice probability is the branch of combinatorics applied to games, simulations, and statistical teaching. The mathematics is elementary but the intuition is often wrong — people systematically underestimate middle sums and overestimate the impact of modifiers. Understanding the actual math makes you better at tabletop games and at explaining probability to anyone. --- What are single die probabilities? For a fair die with N faces (dN), every face has probability 1/N. | Die | Faces | P(each face) | Expected value | Variance | |-----|-------|-------------|----------------|---------| | d4 | 4 | 25% | 2.5 | 1.25 | | d6 | 6 | 16.67% | 3.5 | 2.917 | | d8 | 8 | 12.5% | 4.5 | 5.25 | | d10 | 10 | 10% |…
Frequently Asked Questions
How do I calculate the probability of a dice roll?
For a single die with N faces, each outcome has probability 1/N. For multiple dice, enumerate all combinations (or use the convolution formula) and count how many produce the desired sum. For 2d6 summing to 7: 6 combinations out of 36 total = 6/36 = 16.7%. The number of ways to make sum S with k dice of N faces follows the formula using inclusion-exclusion.
What is the most common sum of 2d6?
7 is the most likely sum of two six-sided dice, with probability 6/36 = 16.67%. This is why 7 is the most rolled number in Craps. The distribution is triangular: sums near the middle (7) have the most combinations, while extremes (2 and 12) have only one combination each.
How do I calculate advantage in D&D?
Advantage in D&D 5e means rolling 2d20 and taking the higher. The probability of getting at least result X with advantage is 1 - ((X-1)/20)^2. Rolling 15 or higher normally is 6/20 = 30%. With advantage: 1 - (14/20)^2 = 1 - 0.49 = 51%. The expected value rises from 10.5 (normal) to approximately 13.8 (advantage).
What is expected value on a d20?
The expected value (average) of a fair d20 is (1+20)/2 = 10.5. For a d6, it is 3.5. For Nd6, the expected value is N × 3.5 (linearity of expectation). For 2d6, average = 7.0. For 3d6, average = 10.5. Expected value is additive regardless of dice type or count.
How do I simulate dice distributions?
Enumerate all possible outcomes by nested loops (practical for small dice counts) or use the polynomial multiplication method: represent each die as a polynomial where x^k has coefficient 1/N for each face k. Multiply polynomials for each die. The coefficient of x^s in the product is the probability of sum s.
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