Dice Probability Calculator (sum of n dice)

Understanding the dice probability calculator — concept and principle

A dice probability calculator gives the exact chance of rolling a particular total with n fair dice of s sides each, and also the chance of a total at or above, or at or below, that target. It is used for board games, tabletop role-playing games (d4, d6, d8, d10, d12, d20), classroom probability lessons and checking game balance.

How it is calculated

The calculator builds the full distribution of the sum by convolution:

1. Start with one way to have a total of 0.
2. For each die, spread the probability of every current total s over s + 1, s + 2, …, s + sides, each with weight 1 ÷ sides.
3. After n dice, the list holds P(sum = t) for every possible t from n to n × sides.

Then P(sum = target) is read directly, P(sum ≥ target) adds all entries from the target upward and P(sum ≤ target) adds all entries from the target downward. Mean sum = n × (sides + 1) ÷ 2. The calculator accepts 1–50 dice with 2–100 sides.

The principle

Each die is independent and every face is equally likely, so every ordered outcome of n dice has probability (1/sides)ⁿ. The probability of a total is the number of outcomes that produce it divided by sidesⁿ. Convolution counts those outcomes efficiently without listing them all. Sums in the middle can be made in many more ways than extreme sums, which is why the distribution peaks at the mean; with several dice it already looks close to a bell curve, an example of the central limit theorem. The variance of the sum is n × (sides² − 1) ÷ 12.

Worked example

1. Inputs: 2 dice, 6 sides, target sum 7.
2. There are 6² = 36 equally likely outcomes. A sum of 7 comes from (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): 6 outcomes.
3. P(sum = 7) = 6 ÷ 36 ≈ 0.1667.
4. Sums 7 to 12 account for 6 + 5 + 4 + 3 + 2 + 1 = 21 outcomes, so P(sum ≥ 7) = 21 ÷ 36 ≈ 0.5833. By symmetry, P(sum ≤ 7) is also 0.5833.
5. Mean sum = 2 × (6 + 1) ÷ 2 = 7.

How to read the result

Seven is the most common total with two six-sided dice, while 2 and 12 each have probability 1/36 ≈ 0.028. Note that "≥ target" and "≤ target" both include the target itself, so the two values add up to 1 + P(sum = target), not to 1. To find the chance of beating a number, use the "≥" value for the next total up.

Common mistakes

• Counting (3,4) and (4,3) as one outcome; the dice are distinguishable, so both count.
• Assuming every total is equally likely.
• Adding the "≥" and "≤" probabilities and expecting 1.

Frequently asked questions

What is the most likely total with three six-sided dice?

Totals 10 and 11 are tied at 27/216 = 0.125 each.

How do I include a modifier such as 2d6 + 3?

Subtract the modifier from the target. A total of 10 on 2d6 + 3 needs 7 on the dice.

Why does the mean equal n × (sides + 1) ÷ 2?

One die averages (1 + sides) ÷ 2, and the mean of a sum is the sum of the means.

Sources: Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1 (Wiley) · Ross, S. M., A First Course in Probability (Pearson)