Dice Probability, Explained

After reading this you can compute the exact chance of rolling any sum with any number of dice, read the shape of the distribution, and avoid the counting mistakes that trip up most people.

What this calculator finds

Roll two ordinary six-sided dice and add the faces. The sum ranges from 2 to 12, but the values are not equally likely. A sum of 7 comes up on 6 of the 36 possible ordered rolls, so its probability is 6/36 \approx 0.1667. A sum of 2 comes up only once, so its probability is 1/36 \approx 0.0278. The calculator gives you the full table of these probabilities for any dice count and any number of sides, plus the chance of landing on a target sum you choose.

The core idea is simple: count the equally likely outcomes that produce your target, then divide by the total number of outcomes. With n dice of s sides each, there are s^n equally likely ordered rolls. For 2 six-sided dice that is 6^2 = 36. For 3 six-sided dice it is 6^3 = 216. The only real work is counting how many of those rolls hit each sum.

When to use it, and when not to

Use this when your dice are fair, independent, and identical, and when you care about the sum. That covers most board games, tabletop role-playing checks against a target number, and school probability problems.

Do not use it for questions about the highest single die, drop-lowest mechanics, or exploding dice. Those change the rule that maps rolls to a number, so the sum distribution no longer applies. Also do not use it when the dice are loaded or linked. The whole method rests on every one of the s^n ordered rolls being equally likely.

The sum of dice is not uniform once you have more than one die. Only a single die gives a flat distribution. Assuming every sum from 2 to 12 is equally likely (probability 1/11) is wrong: 7 is six times as likely as 2.

The formula and the intuition

The clean way to count is the inclusion-exclusion formula. The number of ordered rolls of n dice with s sides that sum to k is:

N(k) = \sum_{j=0}^{\lfloor (k-n)/s \rfloor} (-1)^j \binom{n}{j} \binom{k - s j - 1}{n - 1}

Here n is the number of dice, s is the sides per die, and k is the target sum. The term \binom{n}{j} chooses which dice "overflow" past their maximum, and the alternating sign removes the overcounting. The probability is then N(k)/s^n.

The intuition is easier than the formula. Think of the sum as spreading probability like a bell as you add dice. One die is flat. Two dice pile up in the middle because there are many ways to make a middling sum and few ways to make an extreme one. To reach 2 with two dice you must roll 1 and 1, one way. To reach 7 you can roll 1+6, 2+5, 3+4, 4+3, 5+2, 6+1, six ways.

A faster method for a computer, and the one this calculator uses internally, is convolution. Start with the single-die distribution, then repeatedly combine it with itself. Each combination step adds one more die by summing over every way the new die can fill the gap. It gives the same numbers as the formula above and avoids large factorials.

Worked example: two six-sided dice, target 7

Reproducing the demo

The demo uses n = 2 dice, sides = 6, and target = 7.

Count the total ordered rolls: 6^2 = 36.
Count the rolls that sum to 7. List them: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That is 6 rolls.
Divide: 6/36 = 0.1667, about 16.67%.
Check against the formula. With k=7, n=2, s=6, the upper limit \lfloor (7-2)/6 \rfloor = 0, so only the j=0 term survives: \binom{2}{0}\binom{6}{1} = 6. Same answer.

The sum 7 is the single most likely result with two dice. That is why it anchors so many games.

Exact probabilities for the sum of two six-sided dice
Sum	Ways	Probability	Percent
2	1	1/36	2.78%
3	2	2/36	5.56%
4	3	3/36	8.33%
5	4	4/36	11.11%
6	5	5/36	13.89%
7	6	6/36	16.67%
8	5	5/36	13.89%
9	4	4/36	11.11%
10	3	3/36	8.33%
11	2	2/36	5.56%
12	1	1/36	2.78%

The triangular shape: probability rises to a peak at 7 and falls off symmetrically.

How the shape changes with more dice

With one die the distribution is flat: each face has probability 1/6 \approx 0.1667. With two dice you get the triangle above. With three or more dice the shape rounds into a bell. This is the central limit theorem showing up in a game: adding independent rolls pushes the sum toward a normal curve.

The mean and spread follow simple rules. Each fair die with s sides has mean (s+1)/2 and variance (s^2-1)/12. For six-sided dice that is a mean of 3.5 and a variance of 35/12 \approx 2.917. Because the dice are independent, both quantities add across dice.

\mu = n \cdot \frac{s+1}{2}, \qquad \sigma = \sqrt{n \cdot \frac{s^2 - 1}{12}}

For three six-sided dice the mean is 3 \times 3.5 = 10.5 and the standard deviation is \sqrt{3 \times 2.917} \approx 2.958. The most likely sums, 10 and 11, each have probability 27/216 = 0.125, while the extremes 3 and 18 sit at 1/216 \approx 0.00463.

Reading and interpreting the results

The calculator reports the probability of exactly the target sum, and it is useful to think about three related questions. Exactly hitting 7 with two dice is 16.67%. Rolling 7 or more is the sum of the probabilities from 7 up: (6+5+4+3+2+1)/36 = 21/36 \approx 0.5833. Rolling 7 or less is (1+2+3+4+5+6)/36 = 21/36 as well, because 7 is counted in both tails.

Convert probabilities to odds when it helps a table decision. A 16.67% chance is odds of 5 to 1 against: five ways to miss 7 for every one way that a specific pair lands it, or more directly 30 losing rolls against 6 winning rolls, which reduces to 5 to 1. For rare targets, express the number as "about 1 in N". The sum 2 is 1 in 36.

Common mistakes

The mistakes cluster around counting.

Treating order as irrelevant. The rolls (2,5) and (5,2) are two distinct outcomes. If you count unordered pairs you undercount 7 as 3 instead of 6 and get the wrong probability. Keep the outcomes ordered so all s^n of them are equally likely.
Assuming a flat distribution. As shown, sums are far from uniform once you have two or more dice.
Off-by-one on the range. The minimum sum is n (all ones), not 0 or 1. The maximum is n \times s. A target outside this range has probability 0.
Mixing "exactly" with "at least". Decide which question you are answering before you read the number.

Related calculators

To count the ways a sum can occur without the sign tricks, the Combinations and Permutations calculator handles the underlying \binom{n}{k} terms. To combine the dice result with other independent events, or to work out an at-least-once chance across many rolls, use the Probability calculator. When you want the mean, variance, and standard deviation of a sum you have already tabulated, the Expected Value calculator does it directly. For a related counting surprise, the Birthday Paradox calculator shows how fast coincidences become likely, and the Bayes' Theorem calculator updates a probability once you have new evidence.

Frequently asked questions

What are the odds of rolling a 7 with two dice?

Six of the 36 ordered rolls sum to 7, so the probability is 6/36 \approx 0.1667, or 16.67%. In odds terms that is 5 to 1 against.

Why is 7 the most common sum?

Seven sits in the middle of the range, and the middle has the most combinations. There are six ways to make 7 but only one way each to make 2 or 12. The number of combinations rises toward the center and falls toward the extremes.

How do I find the chance of rolling at least a target?

Add the probabilities for the target and every higher sum. For 7 or more with two dice, that is 21/36 \approx 0.5833. The calculator gives you each exact probability so you can add the ones you need.

Does this work for 20-sided or 4-sided dice?

Yes. Set the sides field to any value. A single d20 is flat at 1/20 = 0.05 per face. Two d20 peak at a sum of 21 with probability 20/400 = 0.05.

Can I mix dice of different sizes?

This calculator assumes identical dice. For mixed pools, such as one d6 and one d8, you would convolve the two different single-die distributions. The method is the same, but the equal-outcome count s^n no longer applies as a single power.

Try it yourself in the tool