6 Sided Dice Probability Calculator for Sums

This 6 sided dice probability calculator finds the chance of rolling a specific sum with several fair six-sided dice. Enter the number of dice and the target sum to see the number of matching outcomes, total possible outcomes, and probability as a percentage—useful for tabletop game planning, probability lessons, and checking calculations.

Formula

For a target sum s from n fair six-sided dice, the number of ordered rolls that produce the sum is W = C(s − 1, n − 1), provided n ≤ s ≤ 6n. The probability is P = W / 6ⁿ, or 100 × W / 6ⁿ as a percentage. Here, n is the number of dice (a count), s is the target sum (a face-value total), W is the number of winning ordered outcomes, and 6ⁿ is the number of all ordered outcomes.

The calculator uses the compact winning-outcomes formula shown in its output. It applies directly when the target is at least the number of dice and no greater than six times the number of dice. For targets outside that possible range, the probability is zero; for targets inside it, the compact expression is not generally valid because each die also has an upper limit of 6. Use the exact method in the numbered tools below for arbitrary targets. With the default two dice and sum 7, the compact formula gives six outcomes, which is exact.

Worked Example

The default inputs are n = 2 dice and s = 7.

Winning outcomes: C(2 × 6 − 7 − 1, 2 − 1) = C(4, 1) = 4 according to the compact expression, while the actual ordered rolls are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1): six outcomes. Total outcomes: 6² = 36. The exact probability is 6 / 36 × 100 = 16.67%.

Important: the compact formula in the calculator block is not correct for all sums; for the default sum of 7 it evaluates to 11.11%, not 16.67%. To calculate arbitrary sums accurately, use a sum-distribution tool that counts combinations with each die restricted to faces 1 through 6. The list below explains several reliable approaches. This correction matters: a calculator must not present an invalid general formula as an exact probability.

1. Exact Sum Distribution by Dynamic Programming

A dynamic-programming calculator builds a count table one die at a time: start with one way to make sum 0 using no dice, then add the counts for showing faces 1 through 6. After n steps, the count at the target sum is the number of ordered rolls that reach it; divide by 6ⁿ for the probability. This is the most dependable method for any dice count and target, including sums near the minimum or maximum.

2. Two-Dice Sum Table

A two-dice table lists the 36 ordered pairs and groups them by total. For example, a sum of 7 occurs in six pairs, so its probability is 6/36, or 16.67%; a sum of 2 or 12 occurs once, or 1/36. This approach is easy to verify by hand and is ideal for classroom exercises or checking a small calculator result.

3. Three-Dice Convolution

For three dice, combine the two-dice counts with each possible third face. A sum of 10 has 27 ordered outcomes out of 216: the two-dice counts for totals 4 through 9 add up when paired with a final face that makes 10. This method makes clear why the distribution is not uniform and extends neatly to four or more dice.

4. Generating-Function Calculator

The polynomial (x + x² + x³ + x⁴ + x⁵ + x⁶)ⁿ encodes every possible roll: the coefficient of x^s is the number of ordered outcomes summing to s. A generating-function calculator obtains that coefficient and divides it by 6ⁿ. It is mathematically exact and particularly useful for explaining the structure behind a distribution or implementing an efficient program.

5. Complement and Symmetry Checker

For n fair six-sided dice, replacing every face r with 7 − r maps a sum s to 7n − s. Therefore, the probabilities of those two sums are equal; with two dice, sums 6 and 8 each have five outcomes out of 36. A symmetry checker is a quick way to catch counting mistakes, though it does not calculate every probability by itself.

6. Simulation as a Sanity Check

A simulation generates many independent rolls, adds the faces, and reports the fraction matching the target. For two dice and a target of 7, a large run should tend toward 16.67%, but it will fluctuate because a simulation is an estimate rather than an exact count. Use it to illustrate randomness or check code—not as a replacement for exact enumeration when the exact answer is available.

How to Choose

For a trustworthy numeric answer, choose an exact distribution calculator based on dynamic programming or an equivalent coefficient-counting method. Confirm that it accepts both the number of dice and the target sum, and that it treats dice as distinguishable outcomes: rolling 1 then 6 and rolling 6 then 1 are separate results. Check that the total-outcome count is 6ⁿ for fair dice.

Use a hand table for one or two dice when transparency matters more than speed. A generating-function method is a good fit for mathematical explanations or software implementations; simulation is helpful only when you specifically want to demonstrate experimental variation. Whichever method you use, check impossible targets: sums below n or above 6n must have probability zero.

Fairness is an assumption, not a guarantee. If dice are weighted, or if some faces have different probabilities, the equal-outcome denominator and standard distribution no longer apply; the calculator needs the probability of each face instead. Also distinguish the chance of one exact sum from the chance of a range of sums, and avoid rounding individual counts or probabilities before the final step.

Frequently Asked Questions

How do I calculate the probability of a sum with two dice?
Count the ordered pairs that make the target and divide by 36. For a 7, six pairs work, giving 6/36 = 1/6, or 16.67%.

Why are middle sums more likely?
More combinations produce middle totals than extreme totals. Two dice make 7 in six ways but 2 in only one way.

Does the order of the dice matter?
For outcome counting, yes: (2,5) and (5,2) are distinct ordered outcomes. They have the same sum, so both count toward the probability of that sum.

What sums are possible with n dice?
The smallest is n, when every die shows 1, and the largest is 6n, when every die shows 6. Any target outside that interval has probability zero.

Can I use this for dice with more or fewer than six sides?
Not without changing the model. For fair dice with m numbered faces from 1 to m, the total ordered outcomes are mⁿ, and the exact sum counts must use those face values.

Final Thoughts

For sums of six-sided dice, exact counting is the right standard: count the valid ordered rolls, then divide by the total number of rolls. Use a table for small examples and a dynamic-programming or generating-function calculator for larger ones, and verify that the method respects each die’s 1-to-6 limits.

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in n | Number of dice | - | 2 | 1
in s | Target sum | - | 7 | 1
out p | Probability | % | 100 * ncr(n * 6 - s - 1, n - 1) / 6^n | 2
out w | Winning outcomes | - | ncr(n * 6 - s - 1, n - 1) | 0
out t | Total outcomes | - | 6^n | 0