What formula is used to calculate dice probabilities?

The tool uses dynamic programming to build the exact probability distribution. Starting with one die whose face probabilities are each 1/sides, it iterates die by die and convolves the distributions, so the final map gives the exact probability of every possible sum. No approximation or simulation is involved.

What is the probability of rolling 7 or more on 2d6?

There are 36 equally likely outcomes when rolling 2d6. Sums of 7-12 cover 21 of those outcomes, so P(sum ≥ 7) = 21/36 ≈ 58.33%. Select "at least" mode with target 7 on a 2d6 configuration to confirm this.

Why does the 2d6 distribution look like a triangle (bell curve)?

Each face of a single d6 has equal probability (1/6). When you add two dice, the middle sums (6, 7, 8) can be formed in many more ways than extreme sums (2 or 12). Sum 7 has 6 combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), while sum 2 has only 1 (1+1). This creates a triangular distribution that approximates a bell curve as more dice are added.

How many dice can the calculator handle?

The tool supports up to 20 dice to keep computation fast in the browser. With 20d6 there are 101 possible sums (20-120) and the distribution is very smooth. For more dice, the Central Limit Theorem means the distribution converges toward a normal curve and you can use the mean and standard deviation shown below the result.

What is the average (mean) sum for a pool of dice?

The mean of a single die with s sides is (s + 1) / 2. For n dice it scales linearly to n(s + 1) / 2. For example, the mean of 3d6 is 3 times (6+1)/2 = 10.5. The calculator always shows the exact computed mean, standard deviation, and most likely sum.

How are custom-sided dice handled?

Type any number from 2 to 100 into the sides input. The tool assumes a fair die with faces numbered 1 through that number. This covers unusual dice like a d3 (often a d6 with faces relabelled 1-1-2-2-3-3), a d7, a d16, or any other symmetric polyhedral or numbered spinner.

Dice Probability Calculator

Every tabletop role-playing game, board game, and wargame comes down to dice rolls — and knowing the exact probability of a roll before you take the risk is a genuine advantage. This dice probability calculator gives you the precise probability distribution for any pool of up to 20 polyhedral dice, standard or custom, in seconds.

How the maths works

The calculator uses dynamic programming to compute the exact probability distribution. The process unfolds in layers:

Start with a single die: every face from 1 to s has probability 1/s.
Add the second die by convolving the current distribution with the single-die distribution — each current sum merges with each face value, multiplying probabilities.
Repeat for each additional die.

The result is a map from every achievable sum to its exact probability. No Monte Carlo simulation, no rounding until display: every figure is mathematically exact for the given configuration.

For n dice each with s sides, the minimum sum is n (all ones) and the maximum is n × s (all maxima). The mean is:

μ = n × (s + 1) / 2

The variance is:

σ² = n × (s² − 1) / 12

These closed-form values match what the dynamic programming engine computes.

Worked example — the classic 2d6

A standard board game or tabletop RPG roll of 2d6 produces sums from 2 to 12. There are 36 equally likely face combinations (6 × 6):

Sum	Combinations	Probability
2	1	2.78%
7	6	16.67%
12	1	2.78%

The mean is (2 + 12) / 2 = 7, confirming the formula μ = 2 × (6 + 1) / 2 = 7. The probability of rolling at least 10 (a common difficulty threshold) is P(sum ≥ 10) = (3 + 2 + 1) / 36 ≈ 16.67%. Change the query to “at least 7” and the probability jumps to 58.33%.

Highlighted use cases

D&D 5e ability score generation — rolling 4d6 and dropping the lowest is not directly modelled here (this tool sums all dice), but you can use it to check the straight 3d6 distribution: the mean is 10.5 with standard deviation ~2.96, and the probability of rolling 15 or higher is about 9.26%.

Warhammer / skirmish games — many games use pools of d6 where you count successes (rolls of 4+, 5+, 6). Select the “at most” mode for failure counts, or “at least” for success counts, on a single die to find the per-die probability, then use the related Binomial Probability Calculator for the full success-count distribution.

Custom polyhedral dice — Dungeon World uses 2d6, many OSR games use a d30 for special tables, some games use d10 pools. Just type the custom number of sides.

Formula note

The core recurrence is:

P(sum = t after die i) = Σ P(sum = t − f after die i − 1) × (1/s)

for f from 1 to s. This is the discrete convolution of uniform distributions, iterated n times. The resulting distribution is a discrete uniform convolution, sometimes called the Irwin-Hall distribution rescaled to integer faces.