Home › Math & Conversions › Core Math & Algebra › Dice roll probability Dice Roll Probability Calculator Compute the probability of sums and successes when rolling one or many dice. Supports exact, at least, at most, and shows odds, percentages, and step-by-step explanations. Core Math & Algebra Interactive dice probability engine Choose whether you care about the sum of the dice or the number of successes (for example “roll 5+ on each die”), then enter your scenario. Sum of dice Number of successes Number of dice 1–15 recommended for exact sums. Sides per die 6 for a standard d6, 20 for a d20, etc. Decimal places 2 decimals 4 decimals 6 decimals Sum condition Type Exactly equal to At least At most Target sum Between min = dice × 1 and max = dice × sides. Show full sum distribution (for small dice pools) Success condition “Success” faces per die For a d6, success on 6 ⇒ 1 face; on 5 or 6 ⇒ 2 faces. Number of successes Exactly At least At most Target successes (k) For “at least one success” set k = 1 and type = “At least”. Calculate probability Clear Example: 2d6 sum = 7 Example: at least one 6 on 4d6 Results Main probability Details Odds & checks Sum distribution Distribution is exact for fair dice. For large pools, only the requested probability is shown to keep things fast. Dice roll probability in a nutshell When you roll fair dice, each face is equally likely. For one s-sided die, the probability of any particular face is \(\dfrac{1}{s}\). For multiple dice, the number of possible outcomes grows quickly: rolling \(n\) independent s-sided dice gives \(s^n\) equally likely outcomes. To get the probability of an event (such as “sum equals 7” or “at least one 6”), you count how many outcomes make the event happen and divide by \(s^n\). Sum of n dice: exact probabilities For \(n\) fair s-sided dice, the sum \(S\) can range from \(n\) (all 1s) up to \(n\,s\) (all s). The probability that the sum equals a specific value \(t\) is \(\mathbb{P}(S = t) = \dfrac{\text{number of combinations of faces that add to } t}{s^n}\). For small \(n\), you can count by hand or via a dynamic programming table that accumulates the number of ways to reach each sum as you add dice one by one. This is exactly what the calculator does internally. Example: 2d6 sum of 7 Two standard six-sided dice (2d6) have \(6^2 = 36\) outcomes. The pairs that sum to 7 are: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\) – six outcomes. \(\mathbb{P}(S = 7) = \dfrac{6}{36} = \dfrac{1}{6} \approx 16.67\%\). At least

Subcategories in Home › Math & Conversions › Core Math & Algebra › Dice roll probability Dice Roll Probability Calculator Compute the probability of sums and successes when rolling one or many dice. Supports exact, at least, at most, and shows odds, percentages, and step-by-step explanations. Core Math & Algebra Interactive dice probability engine Choose whether you care about the sum of the dice or the number of successes (for example “roll 5+ on each die”), then enter your scenario. Sum of dice Number of successes Number of dice 1–15 recommended for exact sums. Sides per die 6 for a standard d6, 20 for a d20, etc. Decimal places 2 decimals 4 decimals 6 decimals Sum condition Type Exactly equal to At least At most Target sum Between min = dice × 1 and max = dice × sides. Show full sum distribution (for small dice pools) Success condition “Success” faces per die For a d6, success on 6 ⇒ 1 face; on 5 or 6 ⇒ 2 faces. Number of successes Exactly At least At most Target successes (k) For “at least one success” set k = 1 and type = “At least”. Calculate probability Clear Example: 2d6 sum = 7 Example: at least one 6 on 4d6 Results Main probability Details Odds & checks Sum distribution Distribution is exact for fair dice. For large pools, only the requested probability is shown to keep things fast. Dice roll probability in a nutshell When you roll fair dice, each face is equally likely. For one s-sided die, the probability of any particular face is \(\dfrac{1}{s}\). For multiple dice, the number of possible outcomes grows quickly: rolling \(n\) independent s-sided dice gives \(s^n\) equally likely outcomes. To get the probability of an event (such as “sum equals 7” or “at least one 6”), you count how many outcomes make the event happen and divide by \(s^n\). Sum of n dice: exact probabilities For \(n\) fair s-sided dice, the sum \(S\) can range from \(n\) (all 1s) up to \(n\,s\) (all s). The probability that the sum equals a specific value \(t\) is \(\mathbb{P}(S = t) = \dfrac{\text{number of combinations of faces that add to } t}{s^n}\). For small \(n\), you can count by hand or via a dynamic programming table that accumulates the number of ways to reach each sum as you add dice one by one. This is exactly what the calculator does internally. Example: 2d6 sum of 7 Two standard six-sided dice (2d6) have \(6^2 = 36\) outcomes. The pairs that sum to 7 are: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\) – six outcomes. \(\mathbb{P}(S = 7) = \dfrac{6}{36} = \dfrac{1}{6} \approx 16.67\%\). At least.

at most a given sum Sometimes you care about “sum at least 10” or “sum at most 4”. In that case: \(\mathbb{P}(S \le t) = \sum_{k=n}^{t} \mathbb{P}(S = k)\) \(\mathbb{P}(S \ge t) = \sum_{k=t}^{n s} \mathbb{P}(S = k)\) The calculator automatically sums over the relevant range for you and reports a single probability. Successes on dice and the binomial distribution Many games define a success as rolling certain faces (e.g. “5 or 6 on a d6”). If there are \(f\) success faces on an s-sided die, the per-die success probability is \(p = \dfrac{f}{s}\). If you roll \(n\) dice, each with success probability \(p\), the number of successes \(K\) follows a binomial distribution : \(\mathbb{P}(K = k) = \binom{n}{k} p^k (1 - p)^{n-k}\). Example: at least one 6 on 4d6 For a d6, one face is a 6, so \(p = 1/6\). For 4 dice, the probability of at least one 6 is \(\mathbb{P}(K \ge 1) = 1 - \mathbb{P}(K = 0) = 1 - (1 - p)^4 = 1 - (5/6)^4 \approx 51.77\%\). Exact probability vs odds It’s often helpful to express probability in different ways: Decimal: a number between 0 and 1, like 0.25. Percentage: the decimal times 100%, like 25%. Odds “1 in N”: \(N \approx 1

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