acalculator

What are the odds of my dice roll?

Pick how many dice you roll, how many sides they have, and a target sum to get the exact chance of rolling it.

Your numbers

Common dice: 4, 6, 8, 10, 12, 20, 100.
The sum is
Probability
0.166667

Exactly 7 on 2 dice with 6 sides: 0.166667 (1/6).

As a fraction
1/6
About 1 in
6
Favourable rolls
6
All possible rolls
36
Average sum
7
Standard deviation of the sum
2.415229

Probability: 0.166667. Exactly 7 on 2 dice with 6 sides: 0.166667 (1/6).

How likely is each sum?

How to calculate

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.

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).

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.

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Number of dice 2, Sides on each die 6, The sum is Exactly, Target sum 7 gives Probability 0.166667, As a fraction 1/6, Favourable rolls 6, All possible rolls 36, About 1 in 6, Average sum 7, Standard deviation of the sum 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. Number of dice 2, Sides on each die 6, The sum is At most, Target sum 10 gives Probability 0.916667, Favourable rolls 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. Number of dice 3, Sides on each die 6, The sum is Exactly, Target sum 10 gives Probability 0.125, As a fraction 1/8, Favourable rolls 27, All possible rolls 216, Average sum 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. Number of dice 1, Sides on each die 20, The sum is At least, Target sum 15 gives Probability 0.3, Favourable rolls 6, About 1 in 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. Number of dice 10, Sides on each die 6, The sum is Exactly, Target sum 35 gives Probability 0.072693, Favourable rolls 4,395,456, All possible rolls 60,466,176, Average sum 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. Number of dice 10, Sides on each die 100, The sum is At least, Target sum 1,000 gives Probability 0, As a fraction 1/100000000000000000000, Favourable rolls 1, All possible rolls 100000000000000000000.Source: Grinstead and Snell, Introduction to Probability, 1997, §7.1 Sums of Discrete Random Variables. https://math.dartmouth.edu/~prob/prob/prob.pdf

How it works

You roll n fair dice, each with s faces numbered 1 to s. Each of the sⁿ possible rolls is equally likely.

Counting the rolls for each sum. Let W₀(0) = 1. For each die added, the number of rolls that give sum k is

Wⱼ(k) = Wⱼ₋₁(k − 1) + Wⱼ₋₁(k − 2) + … + Wⱼ₋₁(k − s),

counting any term with a negative or missing sum as 0. After n dice, Wₙ(k) is the number of rolls whose faces add up to k, for k from n to n × s. All counts are exact whole numbers.

The results:

  • Favourable rolls: the sum of Wₙ(k) over the sums that meet the condition: k equal to the target (exactly), k ≥ target (at least), or k ≤ target (at most).
  • All possible rolls: sⁿ.
  • Probability: favourable ÷ sⁿ, as a decimal.
  • As a fraction: the same ratio in lowest terms, written a/b with no thousands separators, such as 1/6 or 1/100000000000000000000. A probability of 0 shows as 0 and a probability of 1 as 1.
  • About 1 in: sⁿ ÷ favourable, shown to 2 decimal places when there is at least one favourable roll.
  • Average sum: n × (s + 1) ÷ 2.
  • Standard deviation of the sum: √(n × (s² − 1) ÷ 12).

Rules

  • 1 to 10 dice, 2 to 100 sides each, and a whole-number target from 0 to 1,000.
  • A target outside the possible sums is allowed: the probability is then 0 or 1. For example, "at most 1" with 2 dice is 0, and "at least 2" is 1.
  • The decimal probability is the exact fraction rounded once to the nearest 64-bit float, and it shows up to 6 decimal places (6 significant digits below 0.0001), with halves rounded up. Counts show every digit, with thousands separators.

Worked examples by hand

Two dice, exactly 7. The 36 rolls include six with sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Probability = 6/36 = 1/6 ≈ 0.166667, about 1 in 6. Average sum = 2 × 7 ÷ 2 = 7; standard deviation = √(2 × 35 ÷ 12) = 2.415229.

Two dice, at most 10. Only (5,6), (6,5), and (6,6) add up to more than 10, so 36 − 3 = 33 rolls qualify: 33/36 = 11/12 ≈ 0.916667.

Three dice, exactly 10. Counting the rolls: the sums 3 to 10 have 1, 3, 6, 10, 15, 21, 25, 27 rolls, so 10 has 27 of 6³ = 216. Probability = 27/216 = 1/8 = 0.125. Average sum = 3 × 3.5 = 10.5.

One d20, at least 15. Faces 15 to 20 are 6 of 20: probability 0.3, about 1 in 3.33.

Ten dice, exactly 35. The counting rule gives W₁₀(35) = 4,395,456 of 6¹⁰ = 60,466,176 rolls: probability ≈ 0.072693. It is the most likely sum, and equal to the average 10 × 3.5 = 35.

Ten d100 dice, at least 1,000. Only the roll with every die on 100 qualifies: 1 of 100¹⁰ = 10²⁰, a probability of 1 × 10⁻²⁰ (shown as 0.00000000000000000001).

Other questions people ask

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.