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Dice Probability Calculator

Calculate the probability of rolling a specific total with multiple dice.

Result

Probability
16.667%
Ways to Roll This Sum
6
Total Possible Outcomes
36

Minimum possible sum is 2, maximum is 12.

About the Dice Probability Calculator

Rolling a 7 with two six-sided dice feels like it should be as likely as rolling a 2 or 12, but it isn't - some sums have far more ways to occur than others. Our Dice Probability Calculator shows the exact odds for any dice combination and target sum.

How It Works

The calculator builds a full probability distribution for the sum of your dice by starting with a single die's possible outcomes and repeatedly combining in one additional die at a time, tracking how many ways each possible sum can occur, then reads off the count for your specific target sum against the total number of possible outcomes.

Probability = (ways to roll the target sum) ÷ (sides ^ number of dice)

Formula & Methodology

This works by dynamic programming: starting with a distribution representing a single possible outcome (a sum of zero, before any dice are rolled), it adds one die at a time, and for every already-tracked partial sum it spreads that count across each face the new die could show, building up a full tally of how many combinations produce each possible total. The total number of possible outcomes is simply the number of sides raised to the power of the number of dice, since each die's result is independent of the others - dividing the target sum's tally by that total gives the exact probability.

Step-by-Step: Calculating It By Hand

  1. 1Start with a distribution representing zero dice rolled (one way to reach a sum of zero).
  2. 2For each die added, spread every existing partial sum across all possible face values of that die, building an updated distribution.
  3. 3Repeat until all dice have been added, producing the count of ways to reach every possible total sum.
  4. 4Divide the count for your target sum by the total possible outcomes (sides raised to the number of dice) to find the probability.

Examples

Classic two-dice roll

Rolling two standard six-sided dice and targeting a sum of 7 gives a probability of about 16.67%, the highest probability of any sum in that range, since there are 6 ways to make 7 out of 36 total outcomes.

Extreme sum

Targeting a sum of 2 with the same two dice gives only about 2.78% probability, since there's just 1 way (rolling 1 and 1) out of the same 36 total outcomes.

Advantages

  • Calculates exact probabilities for any number of dice, sides, and target sum
  • Shows the number of ways to roll the target alongside the total possible outcomes
  • Handles unusual dice (like a d20) just as easily as standard six-sided dice
  • Useful for understanding why certain dice-game outcomes feel more common than others

Common Mistakes

  • Assuming every possible sum is equally likely, when middle sums are almost always more probable than extreme sums
  • Forgetting the minimum possible sum is the number of dice, not zero or one
  • Confusing the number of ways to roll a sum with the actual probability, which also depends on total possible outcomes
  • Not accounting for how quickly total possible outcomes grow as more dice or more sides are added

Edge Cases to Watch For

  • The minimum possible sum equals the number of dice (all showing a 1), and the maximum equals the number of dice times the number of sides (all showing the highest face).
  • A target sum outside that minimum-to-maximum range has zero ways to occur, and the probability is exactly 0%.
  • Sums near the middle of the possible range are always more probable than sums at the extremes, since there are more combinations of dice values that add up to a middle sum.
  • This assumes standard fair dice with numbered faces 1 through the specified number of sides, uniformly and independently rolled.

Common Use Cases

  • Understanding the odds behind tabletop and board game dice mechanics
  • Probability and statistics education using a concrete, tangible example
  • Game design work that depends on specific dice-sum probabilities
  • General curiosity about why some dice totals come up more often than others
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Why is rolling a 7 with two six-sided dice more likely than rolling a 2 or 12?

There are 6 different ways to make 7 with two dice (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) but only 1 way to make 2 (1+1) or 12 (6+6) - more combinations sum to values near the middle of the range, making them more probable.

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Conclusion

Dice probabilities are a classic, intuitive entry point into combinatorics, and the middle-heavy distribution surprises people the first time they see the exact numbers. Our Dice Roller can simulate actual rolls if you want to see this distribution play out in practice.