# What are the odds of my dice roll?

Computes the exact probability that the sum of up to 10 fair dice is exactly, at least, or at most a target, with the number of ways and the average roll.

- Page: https://www.acalculator.org/statistics/dice-probability-calculator
- JSON spec: https://www.acalculator.org/statistics/dice-probability-calculator.json
- Version: fb2d06a7de00

## Default answer

Example with the default inputs (Number of dice 2, Sides on each die 6, The sum is Exactly, Target sum 7): Exactly 7 on 2 dice with 6 sides: 0.166667 (1/6).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| n | Number of dice | How many dice you roll together, from 1 to 10. |
| s | Sides on each die | The number of faces on each die, numbered 1 to that number: 6 for a standard die, 20 for a d20. |
| cmp | The sum is | Whether the sum must equal the target, be at least the target, or be at most the target. |
| t | Target sum | The total of the faces you are counting. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| probability | Probability | The chance that the sum meets the condition, as a decimal from 0 to 1. |
| fraction | As a fraction | The same probability as an exact fraction in lowest terms, such as 1/6. |
| oneIn | About 1 in | The total rolls divided by the favourable rolls: on average, one roll in this many meets the condition. |
| ways | Favourable rolls | The number of equally likely rolls whose sum meets the condition. |
| total | All possible rolls | The number of equally likely rolls: sides to the power of the number of dice. |
| mean | Average sum | The expected value of the sum: dice × (sides + 1) ÷ 2. |
| sd | Standard deviation of the sum | The spread of the sum: √(dice × (sides² − 1) ÷ 12). |

## Method

Count the rolls of n dice with s sides whose faces add up to each sum (the sidesⁿ rolls are equally likely), then probability = favourable rolls ÷ sidesⁿ.

## Assumptions

- Every die is fair, with faces numbered 1 to the number of sides, and the dice are rolled independently.
- The counts are exact whole numbers, so the fraction is exact; the decimal is rounded once.

## Worked examples

1. n = 2, s = 6, cmp = exactly, t = 7 gives probability = 0.166667, fraction = 1/6, ways = 6, total = 36, oneIn = 6, mean = 7, sd = 2.415229. Source: OpenStax, Contemporary Mathematics, §7.5 Basic Concepts of Probability. https://openstax.org/books/contemporary-mathematics/pages/7-5-basic-concepts-of-probability.
2. n = 2, s = 6, cmp = atMost, t = 10 gives probability = 0.916667, ways = 33. Source: OpenStax, Contemporary Mathematics, §7.5 Basic Concepts of Probability. https://openstax.org/books/contemporary-mathematics/pages/7-5-basic-concepts-of-probability, Example 7.22 (1 − 3/36 = 11/12).
3. n = 3, s = 6, cmp = exactly, t = 10 gives probability = 0.125, fraction = 1/8, ways = 27, total = 216, mean = 10.5. Source: Grinstead and Snell, Introduction to Probability, 1997, §7.1 Sums of Discrete Random Variables. https://math.dartmouth.edu/~prob/prob/prob.pdf.
4. n = 1, s = 20, cmp = atLeast, t = 15 gives probability = 0.3, ways = 6, oneIn = 3.333333. Source: OpenStax, Contemporary Mathematics, §7.5 Basic Concepts of Probability. https://openstax.org/books/contemporary-mathematics/pages/7-5-basic-concepts-of-probability.
5. n = 10, s = 6, cmp = exactly, t = 35 gives probability = 0.072693, ways = 4,395,456, total = 60,466,176, mean = 35. Source: Grinstead and Snell, Introduction to Probability, 1997, §7.1 Sums of Discrete Random Variables. https://math.dartmouth.edu/~prob/prob/prob.pdf.
6. n = 10, s = 100, cmp = atLeast, t = 1,000 gives probability = 0, fraction = 1/100000000000000000000, ways = 1, total = 100000000000000000000. Source: Grinstead and Snell, Introduction to Probability, 1997, §7.1 Sums of Discrete Random Variables. https://math.dartmouth.edu/~prob/prob/prob.pdf.

## FAQ

### What is the probability of rolling a 7 with two dice?

Two six-sided dice have 6 × 6 = 36 equally likely rolls, and 6 of them add up to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So the probability is 6/36 = 1/6, about 0.1667. Seven is the most likely sum of two dice.

### How is the probability of a dice sum calculated?

The calculator counts how many of the equally likely rolls give each sum. It starts with one die (each face once), then adds one die at a time: the count for a sum k is the total of the previous counts for k − 1, k − 2, …, k − sides. The probability is the count of rolls that meet your condition divided by sidesⁿ.

### Why is the middle sum the most likely?

More combinations of faces add up to a middle value than to an extreme one. Two dice make 2 only as (1,1), but 7 in six ways. With more dice the counts form a bell shape around the average sum.

### What is the average roll of several dice?

One die with s sides averages (s + 1) ÷ 2, so a d6 averages 3.5 and a d20 averages 10.5. For n dice multiply by n: 3d6 averages 10.5 and 2d10 averages 11.

### What are the odds of rolling 15 or more on a d20?

Six faces (15 to 20) of 20 meet it, so the probability is 6/20 = 0.3, or about 1 roll in 3.33.

### Does the calculator handle dice like 2d6 or 3d8?

Yes. 2d6 means two dice with six sides; enter 2 dice and 6 sides. All dice in one roll have the same number of sides, from 2 to 100, and you can roll up to 10 of them.

## Sources

- OpenStax, Contemporary Mathematics, §7.5 Basic Concepts of Probability (two dice, 36 outcomes). https://openstax.org/books/contemporary-mathematics/pages/7-5-basic-concepts-of-probability
- Grinstead and Snell, Introduction to Probability, 1997, §7.1 Sums of Discrete Random Variables. https://math.dartmouth.edu/~prob/prob/prob.pdf