What is the expected value (average) when rolling dice?

The expected value of one die with s sides is (1 + s) / 2. For a d6 that is 3.5. For n dice the expected value multiplies: n x (1 + s) / 2. For 2d6 it is 2 x 3.5 = 7. For 3d6 it is 3 x 3.5 = 10.5. The expected value is the long-run average sum if you rolled many times.

Dice Probability Calculator

Q: What is the probability of rolling a natural 20 on a d20?

A natural 20 (the maximum result on a single d20) has probability 1/20 = 5%. In D&D this is called a critical hit. If you need to roll a 20 on a d20, each roll has exactly a 5% chance of success. With advantage (rolling twice and taking the higher), the probability of getting at least one 20 = 1 - (19/20)^2 = 1 - 0.9025 = 9.75%.

Q: What is the probability of rolling at least 8 on 2d6?

Use the range mode with minimum 8 and maximum 12. There are 15 ways to roll 8 or higher with 2d6 (out of 36). Ways: sum 8 = 5, sum 9 = 4, sum 10 = 3, sum 11 = 2, sum 12 = 1. Total = 15. Probability = 15/36 = 41.67%.

Q: What is the probability distribution shape for multiple dice?

For a single die, the distribution is uniform (each face equally likely). For two or more dice, the distribution of sums is triangular (for 2 dice) or bell-shaped (for 3+ dice). This is because there are more ways to achieve middle sums than extreme sums. As you add more dice, by the Central Limit Theorem, the distribution approaches a normal (Gaussian) curve.

Find the exact probability of any dice roll outcome. Supports standard d6, d8, d10, d12, d20 and custom dice.

🎲 What is Dice Probability?

Dice probability is the branch of combinatorics and probability theory that deals with computing the likelihood of specific outcomes when one or more dice are rolled. Each face of a fair die is equally likely to appear, making dice problems elegant examples of discrete uniform distributions and, when multiple dice are combined, of convolution in probability.

The most familiar example is rolling two six-sided dice (2d6) and asking for the probability of a given sum. The total number of equally likely outcomes is 6 × 6 = 36. Because there are more ways to achieve a middle sum (like 7) than an extreme one (like 2 or 12), the distribution is triangular rather than uniform. The sum 7 can be made in 6 ways out of 36, giving a probability of 1/6 ≈ 16.67%.

Dice probability has practical applications in board games, tabletop role-playing games (such as Dungeons & Dragons), war games, casino games, and in teaching statistics and probability. It also appears in real-world risk modelling, where sums of independent random variables approximate the shape that many dice together produce.

As you roll more dice, the distribution of sums moves toward a bell curve. This is a direct consequence of the Central Limit Theorem: the sum of many independent, identically distributed random variables converges to a normal distribution regardless of the shape of the original distribution. Even with just 3 or 4 dice, the bell-shaped pattern is clearly visible.

This calculator uses a dynamic programming algorithm to compute exact probabilities for up to 10 dice each with up to 20 sides. The results are shown as a percentage, a simplified fraction, the raw count of favourable outcomes, and the total number of equally likely outcomes.

Formula and Algorithm

For n dice each with s sides, the total number of equally likely outcomes is:

Total outcomes = sⁿ

The number of ways to achieve a specific sum k is the count of ordered tuples (d1, d2, ..., dn) where each di is between 1 and s and d1 + d2 + ... + dn = k. This count is computed using dynamic programming:

dp[0][0] = 1 → dp[d][s] = ∑ dp[d-1][s-f] for f = 1 to sides

dp[d][s] = number of ways to achieve sum s with d dice

f = face value (1 through sides) of the current die

The probability of rolling exactly the target sum k is:

P(X = k) = dp[n][k] / sⁿ

For a range [min, max], sum the counts across all sums in the range:

P(min ≤ X ≤ max) = ∑ dp[n][k] for k = min to max, divided by sⁿ

Expected value of the sum of n dice each with s sides:

E[sum] = n × (s + 1) / 2

For 2d6: E = 2 × (6 + 1) / 2 = 7

For 4d6: E = 4 × 3.5 = 14

How to Use This Calculator

Choose your mode — select Exact Sum to find the probability of rolling a specific total, or Sum Range to find the probability of rolling within minimum and maximum bounds.
Set number of dice — enter how many dice you are rolling (1 to 10).
Set sides per die — enter the number of faces on each die (2 to 20). Common values: 4 (d4), 6 (d6), 8 (d8), 10 (d10), 12 (d12), 20 (d20).
Enter your target — for Exact mode, enter the sum you want to hit. For Range mode, enter the minimum and maximum sums (both inclusive).
Click Calculate — view the probability as a percentage, simplified fraction, favourable ways, and total outcomes.

Example Calculations

Example 1 — Most Common 2d6 Sum

Probability of rolling a sum of 7 with 2 standard six-sided dice

Total outcomes = 6² = 36

Ways to sum to 7: (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) = 6 ways

P = 6/36 = 1/6 ≈ 16.67%

Probability = 16.6667% · Fraction = 1/6 · 6 ways out of 36

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Example 2 — Three Dice Sum

Probability of rolling a sum of 10 with 3d6

Total outcomes = 6³ = 216

Ways to sum to 10 with 3d6 = 27 (computed by DP)

P = 27/216 = 1/8 = 12.50%

Probability = 12.50% · Fraction = 1/8 · 27 ways out of 216

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Example 3 — Range Probability (2d6 sum at least 8)

Probability of rolling 8 or higher (8 through 12) with 2d6

Ways: sum 8=5, sum 9=4, sum 10=3, sum 11=2, sum 12=1. Total = 15

P = 15/36 ≈ 41.67%

Probability = 41.6667% · 15 ways out of 36

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Example 4 — Natural 20 on a d20

Probability of rolling the maximum (20) on a single d20

Total outcomes = 20¹ = 20

Ways to roll 20 = 1

P = 1/20 = 5.00%

Probability = 5.0000% · Fraction = 1/20 · 1 way out of 20

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Frequently Asked Questions

What is the probability of rolling a 7 with two six-sided dice?+

With 2d6 there are 6 x 6 = 36 equally likely outcomes. The combinations that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - exactly 6 ways. Probability = 6/36 = 1/6 = 16.67%. The number 7 is the most probable sum on 2d6 because it has the greatest number of combinations.

How do you calculate dice roll probability?+

For n dice each with s sides, total outcomes = s^n. To find the probability of a specific sum k, count the number of ways to partition k into n parts each between 1 and s (order matters). This counting is most efficiently done with dynamic programming. Probability = ways / s^n.

What dice are used in Dungeons and Dragons?+

D&D uses seven standard dice: d4, d6, d8, d10, d12, d20, and d100 (two d10s). The d20 is used for most skill checks and attacks. Damage dice vary by weapon and spell. Use this calculator to find probabilities for any of these common dice types by entering the appropriate number of sides.

What is the probability of rolling a natural 20 on a d20?+

A natural 20 on a single d20 has probability 1/20 = 5%. In D&D this is called a critical hit. With advantage (rolling twice and taking the higher result), the probability of at least one 20 = 1 - (19/20)^2 = 9.75%.

How many ways can you roll a sum of 10 with 3d6?+

With 3d6, the total outcomes are 6^3 = 216. The number of ways to sum to 10 with three dice each showing 1-6 is 27. P(sum = 10 with 3d6) = 27/216 = 12.50%. Use this calculator with diceCount=3, diceSides=6, diceTarget=10 to verify.

What is the expected value when rolling dice?+

The expected value of one die with s sides is (1 + s) / 2. For a d6 that is 3.5. For n dice the expected value multiplies: n x (1 + s) / 2. For 2d6 it is 7, for 3d6 it is 10.5, for 4d6 it is 14. The expected value is the long-run average sum if you rolled many times.

What is the probability of rolling at least 8 on 2d6?+

Use the range mode with minimum 8 and maximum 12. Ways: sum 8=5, sum 9=4, sum 10=3, sum 11=2, sum 12=1. Total ways = 15. Probability = 15/36 = 41.67%.

Does the order of dice rolls matter for probability?+

For sum calculations, order matters in counting outcomes. When you roll 2d6 and get a 1 and a 6, there are two distinct outcomes: (1 from die1, 6 from die2) and (6 from die1, 1 from die2). Both count toward the sum of 7. Treating dice as distinguishable gives the correct uniform probability distribution. This calculator uses this standard convention.

What is the probability distribution shape for multiple dice?+

For a single die, the distribution is uniform. For two dice, it is triangular. For three or more dice, it becomes increasingly bell-shaped. This is the Central Limit Theorem at work: the sum of many independent random variables approaches a normal distribution regardless of the original distribution shape.

Can I calculate probabilities for non-standard dice?+

Yes. This calculator supports any number of sides from 2 to 20. You can model d4, d6, d8, d10, d12, d20, or even custom dice like d7, d13, or d15. Many board games and wargames use non-standard dice. Simply enter the number of sides in the Sides per Die field.

Dice Probability Calculator

🎲 What is Dice Probability?

Formula and Algorithm

How to Use This Calculator

Example Calculations

Example 1 — Most Common 2d6 Sum

Probability of rolling a sum of 7 with 2 standard six-sided dice

Example 2 — Three Dice Sum

Probability of rolling a sum of 10 with 3d6

Example 3 — Range Probability (2d6 sum at least 8)

Probability of rolling 8 or higher (8 through 12) with 2d6

Example 4 — Natural 20 on a d20

Probability of rolling the maximum (20) on a single d20

Frequently Asked Questions

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