How to use the dice pool success probability calculator
- Enter the number of dice in the pool and the faces on each die.
- Enter the roll that counts as a success, such as 8 or higher on a ten-sided die.
- Enter how many successes you need and pick a reroll rule if your game has one.
- Read the chance of at least, exactly and at most that many successes, plus the average and the table for every count.
Formula
Chance one die succeeds p = (faces - target + 1) / faces
Reroll misses once: p = p0 + (1 - p0) x p0; reroll 1s once: p = p0 + (1 / faces) x p0
P(exactly k successes) = C(n, k) x p^k x (1 - p)^(n - k)
Mean = n p; standard deviation = sqrt(n p (1 - p))
The dice are independent and fair, so the count of successes follows the binomial distribution. At least k is the sum of the exact chances from k up to n.
Five ten-sided dice, success on 8 or more
Each die succeeds 30 percent of the time. For 5 dice, the chance of at least 2 successes is 47.178 percent, of exactly 2 is 30.87 percent and of at most 2 is 83.692 percent. No successes at all comes up 16.807 percent of the time. The average is 1.5 successes with a standard deviation of 1.0247, and one success is the most likely count.
If misses can be rerolled once, each die succeeds 51 percent of the time and the chance of at least 2 successes rises to 82.475 percent. For 3 six-sided dice counting 5 or 6, the chances of 0, 1, 2 and 3 successes are 64, 96, 48 and 8 in 216.
Pools versus sums
Some games add up the dice, and others count how many dice reach a target. Success-counting pools are common in role-playing games where a skill gives a number of dice, and the question is how many show a hit. Summing is covered by the dice probability calculator. For a chain of rolls such as hit, wound and save use the hit, wound and save calculator.
For counting arrangements see the combinations and permutations calculator, and for distances on a hex map the hex grid calculator.
Reading the table
The table lists the chance of each exact number of successes and of at least that many. The at-least column falls from 100 percent at zero toward 0 at the full pool. Use it to see how likely a target is, or how much one extra die changes your odds.
Assumptions and limits
- Every die is fair and independent, and a success is a roll of the target number or higher.
- A reroll applies once to each die that qualifies, and the reroll counts if it succeeds.
- Exploding dice, canceling failures and automatic successes are not modeled.
- The pool is from 1 to 100 dice with 2 to 100 faces.
Frequently asked questions
How do I calculate the chance of getting a number of successes on dice?
Find the chance one die succeeds, then use the binomial formula for the pool. For 5 dice at 30 percent each, at least two successes happen 47.18 percent of the time.
How does a reroll change the odds?
Rerolling a miss gives each die a second chance, so the success chance goes from p to p + (1 - p) x p. Thirty percent becomes 51 percent.
What is the average number of successes?
It is the number of dice times the chance one succeeds. Five dice at 30 percent average 1.5 successes.
Is the pool the same as rolling and adding?
No. A pool counts how many dice meet a target, while adding uses the total of the faces. They answer different questions and need different formulas.