Calculate dice rolling probabilities for any number of dice across d4, d6, d8, d10, d12, d20, and d100 with exact, at least, and at most sums.
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The Dice Probability Calculator computes the probability of rolling specific sums, target thresholds, and combinations with polyhedral dice. Essential for tabletop roleplaying games (D&D, Pathfinder), board games (Settlers of Catan, Craps), and statistical simulation.
P(S = s) = \frac{1}{d^n} \sum_{k=0}^{\lfloor \frac{s-n}{d} \rfloor} (-1)^k \binom{n}{k} \binom{s - d k - 1}{n - 1}Where n is number of dice, d is sides per die, and s is the target sum. The total number of outcomes is d^n.
Scenario: Find the probability of rolling a total sum of 7 when rolling two standard dice in Craps or Catan.
Total outcomes = 6 × 6 = 36.
Favorable pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 combinations.
Calculate probability: P(Sum = 7) = 6 / 36 = 1/6 ≈ 16.67%.
Odds against rolling a 7: 5 to 1.
When rolling two standard 6-sided dice, 7 is the most probable sum with 6 combinations out of 36 (16.67% probability).
On a fair 20-sided die, every number from 1 to 20 has an equal 1 in 20 chance, or exactly 5.00% probability.
Rolling with advantage (take the higher of two d20s) raises your chance of getting a natural 20 from 5.0% to 9.75% (1 - 0.95² = 0.0975), and significantly boosts your average roll from 10.5 to 13.825.
There are 6 possible doubles (1-1, 2-2, 3-3, 4-4, 5-5, 6-6) out of 36 outcomes, giving a 6/36 = 1/6 ≈ 16.67% probability.
Calculate the probability of single events, union of events, intersection, conditional probability P(A|B), and complements.
Calculate probabilities for coin toss trials, exact heads/tails count, at least k heads, and consecutive streak probabilities.
Compute exact P(X=k) and cumulative binomial probabilities P(X≤k), P(X≥k), distribution mean, variance, and standard deviation.