Dice Pool Probability: How Likely Are a Given Number of Successes?
A dice pool counts individual dice that meet a stated threshold. If 6 fair d10 dice succeed on 8, 9, or 10, one die has probability 3/10 and exactly 2 successes have probability C(6,2)(0.3)^2(0.7)^4 = 32.41%.
A worked example with numbers
Six d10 dice need an 8 or higher. The favourable faces are 8, 9, and 10, so p = 3/10. Exactly two successes can occupy C(6,2) = 15 positions; each arrangement has probability 0.3² × 0.7⁴. The result is 15 × 0.09 × 0.2401 = 0.324135, or 32.41%. At least two successes is larger: add exactly 3 through 6 successes.
How to set the rule before the result
Set the number of dice, faces, success threshold, and whether the question means exactly, at least, or at most before rolling. The dice pool roller displays a success-count rule; a dice probability calculator is better for checking a named event.
Common mistakes that change the odds or the process
Do not add face values when the game counts successes. Do not use p² for “at least two” in six dice: that omits the other four dice and all cases with more successes. Do not quietly treat rerolls, exploding maximums, or doubled successes as ordinary binary dice.
Where this method stops being appropriate
The binomial model requires equal independent dice and one unchanged threshold. It does not describe a system with shared resources, conditional rerolls, loaded dice, or a rule that keeps only the best result.
How the random source fits into the rule
NIST’s binomial-distribution reference
Apply the rule to the actual input
Set the number of dice, faces, success threshold, and whether the question means exactly, at least, or at most before rolling. The dice pool roller displays a success-count rule; a dice probability calculator is better for checking a named event.
Audit the calculation or allocation
Six d10 dice need an 8 or higher. The favourable faces are 8, 9, and 10, so p = 3/10. Exactly two successes can occupy C(6,2) = 15 positions; each arrangement has probability 0.3² × 0.7⁴. The result is 15 × 0.09 × 0.2401 = 0.324135, or 32.41%. At least two successes is larger: add exactly 3 through 6 successes.
Do not import a different rule by accident
Do not add face values when the game counts successes. Do not use p² for “at least two” in six dice: that omits the other four dice and all cases with more successes. Do not quietly treat rerolls, exploding maximums, or doubled successes as ordinary binary dice.
Limit of this specific method
The binomial model requires equal independent dice and one unchanged threshold. It does not describe a system with shared resources, conditional rerolls, loaded dice, or a rule that keeps only the best result.
Source and reproducibility
NIST’s binomial-distribution reference
Build a dice-pool table before using a formula
| Successes in 6 dice | Probability when p = 0.30 |
|---|---|
| 0 | 11.76% |
| 1 | 30.26% |
| 2 | 32.41% |
| 3 | 18.52% |
| 4 or more | 7.06% |
The table adds to 100% after rounding. It separates “exactly two” from “at least two”: the latter is 32.41% + 18.52% + 7.06% = 57.99%. A game rule that awards a bonus at three successes asks a different question from a rule that merely requires two.
Translate special dice rules first
If a maximum face explodes, one die can generate more than one success and the simple six-trial binomial table no longer applies. If 10 counts as two successes, define a three-outcome die: failure, one success, two successes. If a player rerolls failures once, calculate the chance of final failure before counting the pool. Those changes are ordinary game design, but each changes the elementary outcome being counted.
Reproduce this result before relying on it
A dice pool is a group of independent dice in which each die is classified as a success or failure by one announced threshold. Six d10 dice need an 8 or higher. The favourable faces are 8, 9, and 10, so p = 3/10. Exactly two successes can occupy C(6,2) = 15 positions; each arrangement has probability 0.3² × 0.7⁴. The result is 15 × 0.09 × 0.2401 = 0.324135, or 32.41%. At least two successes is larger: add exactly 3 through 6 successes.
Choose the action that matches the stated rule
Set the number of dice, faces, success threshold, and whether the question means exactly, at least, or at most before rolling. The dice pool roller displays a success-count rule; a dice probability calculator is better for checking a named event. Do not add face values when the game counts successes. Do not use p² for “at least two” in six dice: that omits the other four dice and all cases with more successes. Do not quietly treat rerolls, exploding maximums, or doubled successes as ordinary binary dice.
What the number does not decide
The binomial model requires equal independent dice and one unchanged threshold. It does not describe a system with shared resources, conditional rerolls, loaded dice, or a rule that keeps only the best result. NIST’s binomial-distribution reference
State the complete decision model
A dice pool counts individual dice that meet a stated threshold. If 6 fair d10 dice succeed on 8, 9, or 10, one die has probability 3/10 and exactly 2 successes have probability C(6,2)(0.3)^2(0.7)^4 = 32.41%. A dice pool is a group of independent dice in which each die is classified as a success or failure by one announced threshold.
Before publishing or using the outcome
Six d10 dice need an 8 or higher. The favourable faces are 8, 9, and 10, so p = 3/10. Exactly two successes can occupy C(6,2) = 15 positions; each arrangement has probability 0.3² × 0.7⁴. The result is 15 × 0.09 × 0.2401 = 0.324135, or 32.41%. At least two successes is larger: add exactly 3 through 6 successes. Set the number of dice, faces, success threshold, and whether the question means exactly, at least, or at most before rolling. The dice pool roller displays a success-count rule; a dice probability calculator is better for checking a named event.
Final rule check for this allocation
A dice pool counts individual dice that meet a stated threshold. If 6 fair d10 dice succeed on 8, 9, or 10, one die has probability 3/10 and exactly 2 successes have probability C(6,2)(0.3)^2(0.7)^4 = 32.41%. A dice pool is a group of independent dice in which each die is classified as a success or failure by one announced threshold. The binomial model requires equal independent dice and one unchanged threshold. It does not describe a system with shared resources, conditional rerolls, loaded dice, or a rule that keeps only the best result.