How do you calculate probability in dice?

You divide the number of outcomes that satisfy a certain condition by the total number of possible outcomes. With a six-sided die, each side has an equal chance of being rolled so the probability for any single roll is one in six. When using multiple dice, you count up how many combinations are equally likely to produce the desired outcome.

What is the probability of getting a sum of 7 with two dice?

There are 36 different equally likely ways to roll two six-sided dice. Six of those combinations add up to a sum of 7. So the probability is 6/36, or 1/6, which is about 16.67 percent. The sum of 7 is the most common sum when rolling two dice.

What is the probability of rolling at least one six?

To calculate the probability that a six-sided die never rolls a 6, we subtract this from one. The chance of not rolling a 6 on any given roll is five in six. So the chance of rolling at least one 6 in n rolls is one minus five-sixths to the nth power. For four dice, this comes out to about 51.8 percent.

Can it be used with dice that have many or few sides?

Yes. Adjust the number of sides per dice according to which dice you are using e.g. 4, 8, 10, 12 or 20. As the formula uses the number of sides directly, the distribution of sums and probabilities for certain results will automatically update with a fair dice.

What is the expected value of rolling a die?

The expected value is the average of the results you'd get if you rolled the dice many times. The average for a six-sided die is 3.5, so the expected value for the sum of n dice is what you get when you multiply 3.5 by n. This is an average in the long term and doesn't tell you what will happen on any one roll.

6-Sided Dice Probability Calculator

Your roll

What are you rolling for?

The sum of the dice

A chosen value across the dice

Number of dice

Sides per die

The result

Target sum

Probability

You are finding the probability of a total across your dice. Set the comparison and target sum below.

Probability

As a decimal
About 1 in
Odds against (to 1)

Favorable outcomes
Total outcomes

Expected value
Standard deviation

Lowest possible sum
Highest possible sum

That is about a 16.67% chance for the outcome you asked about.

Full distribution

Show the full distribution

Plot every outcome's probability and highlight the ones you asked about.

Every outcome and its probability
Outcome	Ways / probability	Chance
2	1	2.78%
3	2	5.56%
4	3	8.33%
5	4	11.11%
6	5	13.89%
7	6	16.67%
8	5	13.89%
9	4	11.11%
10	3	8.33%
11	2	5.56%
12	1	2.78%

A dice probability calculator is a tool that can be used to calculate the probability of a particular outcome when rolling dice. To use the calculator, you will need to enter the number of dice being rolled and the desired outcome. The calculator will then display the exact probability of that outcome in percent, decimal, and "1 in N" form. There are two common types of calculations that can be performed using a dice probability calculator: calculating the probability of a particular sum when rolling multiple dice, and calculating the probability of a particular number appearing on a given die.

Probability of a single die

Let's start with a fair six-sided die. Since each side is equally likely to come up, the probability of any one number coming up is the same.

P(\text{a chosen face}) = \frac{1}{6} \approx 0.1667 = 16.67\%

The basic calculation for a cube is to divide the number of results that meet a certain condition by the total possible outcomes. The probability of rolling a "4" is one in six since it's only one of six possible outcomes. The probability of rolling an even number is half, as there are three out of six possible outcomes.

Probability of sum of multiple dice

When you increase the number of dice to two, the probabilities for each sum change. Since any side on one die can be combined with any side on the second die, there are a total of 36 possible outcomes. There are six combinations that produce a sum of 7, which is the most common sum. The combinations that produce sums of 2 or 12 are only possible once, making them the least likely.

P(\text{sum} = 7 \text{ with two dice}) = \frac{6}{36} = \frac{1}{6} \approx 16.67\%

The following table shows all possible sums with two six-sided dice, how many of the 36 combinations produce each sum, and the probability for each sum.

Sum	Ways	Probability
2	1	2.78%
3	2	5.56%
4	3	8.33%
5	4	11.11%
6	5	13.89%
7	6	16.67%
8	5	13.89%
9	4	11.11%
10	3	8.33%
11	2	5.56%
12	1	2.78%

The distribution forms a triangle with its peak at 7. When more than three dice are used there are many more combinations that lead to a sum near the average than to extreme sums so the peak becomes a rounded bell curve. This calculator counts the number of combinations exactly and therefore gives correct probabilities regardless of the number of dice.

Probability of a certain number of eyes:

Another common calculation is to find the probability of rolling a certain number (e.g., a 6). To roll a specific number with all the dice at once, that number must be rolled on every individual die. Therefore, the probabilities for each individual die are multiplied together.

P(\text{all } n \text{ dice show the value}) = \left(\frac{1}{6}\right)^{n}

The probability of both dice showing a 6 is 1 in 36. To calculate the probability that an event will happen at least once, it is sometimes easier to calculate the chance that it does not happen at all and then subtracting from 1.

P(\text{at least one}) = 1 - \left(\frac{5}{6}\right)^{n}

The probability of getting at least one six on four dice is the value obtained by subtracting from 1 the fourth power of five sixths, which is approximately 51.8 percent. When the number of dice (e.g., five) and the exact number of desired results (e.g., exactly two 6s) are fixed, then the formula for binomial distribution is used. This calculates the number of combinations in which particular dice match, as well as the probability of a match and the probability of not matching.

P(\text{exactly } k) = \binom{n}{k}\left(\frac{1}{6}\right)^{k}\left(\frac{5}{6}\right)^{n-k}

Expected Value and Variance

The expected value is the long term average if you repeat something often enough. The average of a single six-sided dice roll is 3.5 and it ranges from 1 to 6. So the expected value for the sum of n dice rolls is the result when you multiply n by 3.5. The standard deviation tells how much the typical results deviate from the average. As more dice are used, the standard deviation gets larger but since it grows slower than the sum, the results tend to cluster closer to the average as more dice are used.

E[\text{sum}] = n \cdot \frac{6+1}{2} = 3.5n

The table below shows the input parameters used by this calculator and examples for each entry.

Input	Meaning	Example
n	Number of dice	2
sides	Faces per die	6
target	Sum or count you ask about	7
comparison	Exactly, at least, at most, and so on	exactly

How to use this calculator:

First select whether you want to calculate the sum of dice or how often a particular result occurs. Enter the number of dice and the number of sides per die. For standard dice the number of sides will remain at 6, but it can also be changed to account for d4-, d8-, d10-, d12- and d20-dice. When you select criteria and a target value, the probability is shown in the results columns as a percentage, decimal and in 1-in-N format. In addition, the expected value, variance and full probability distribution are also shown so that you can check all of your results at a glance.

This calculator is for educational and planning purposes only. It uses fair and independent dice. As real dice may have weight differences or not roll evenly, the results are guidelines and not guarantees.

Frequently asked questions

How do you calculate probability in dice?: You divide the number of outcomes that satisfy a certain condition by the total number of possible outcomes. With a six-sided die, each side has an equal chance of being rolled so the probability for any single roll is one in six. When using multiple dice, you count up how many combinations are equally likely to produce the desired outcome.
What is the probability of getting a sum of 7 with two dice?: There are 36 different equally likely ways to roll two six-sided dice. Six of those combinations add up to a sum of 7. So the probability is 6/36, or 1/6, which is about 16.67 percent. The sum of 7 is the most common sum when rolling two dice.
What is the probability of rolling at least one six?: To calculate the probability that a six-sided die never rolls a 6, we subtract this from one. The chance of not rolling a 6 on any given roll is five in six. So the chance of rolling at least one 6 in n rolls is one minus five-sixths to the nth power. For four dice, this comes out to about 51.8 percent.
Can it be used with dice that have many or few sides?: Yes. Adjust the number of sides per dice according to which dice you are using e.g. 4, 8, 10, 12 or 20. As the formula uses the number of sides directly, the distribution of sums and probabilities for certain results will automatically update with a fair dice.
What is the expected value of rolling a die?: The expected value is the average of the results you'd get if you rolled the dice many times. The average for a six-sided die is 3.5, so the expected value for the sum of n dice is what you get when you multiply 3.5 by n. This is an average in the long term and doesn't tell you what will happen on any one roll.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

ThoughtCo: Probabilities for Rolling Two Dice
The 36-combination table and sum probabilities for two dice.
NIST Special Publication 811: Guide for the Use of the International System of Units
The US national standard for SI units, unit names and symbols, and conversion factors.