Probability
Dice Sum Outcome Table Calculator
Enumerate exact sum counts and probabilities for identical fair dice, calculate a selected-sum probability, expected hits, and a benchmark gap.
Minimum possible sum-
Maximum possible sum-
Complete equally likely outcome count-
Ways to selected exact sum-
Selected exact-sum probability-
Selected probability as percentage-
Expected selected outcomes-
Final sum in table-
Selected probability minus entered comparison-
Decision view
Exact dice-sum probability distribution
Exact dice-sum probability distributionEach discrete sum has its own probability bar; adjacent outcomes are not connected as though the result were continuous.
Exact scenario comparisonSelected exact sum changes while all other entered assumptions remain constant.
| Selected exact sum | Minimum possible sum | Maximum possible sum | Complete equally likely outcome count | Ways to selected exact sum | Selected exact-sum probability | Selected probability as percentage | Expected selected outcomes | Final sum in table | Selected probability minus entered comparison |
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Period-by-period detail
Exact dice-sum outcome table
How to use Dice Sum Outcome Table Calculator
- Enter dice count and sides.
- Choose the displayed sum range and selected sum.
- Enter planned rolls and a comparison probability.
- Read the full discrete histogram and exact table.
Calculator guide
Understanding Dice Sum Outcome Table Calculator
Dice sums form a discrete distribution because each total has a specific number of ordered face combinations. Exact enumeration is more informative than treating all totals as equally likely.
Calculation method
How the calculation works
Enumerate exact sum combinations for every displayed total from identical fair dice and divide each by the complete outcome space. Count the complete outcome space as s^d, count ordered outcomes producing each sum, and divide favorable ways by total outcomes.
Detailed calculation process
Enumerate the exact dice-sum distribution
The default uses two six-sided dice, displays sums 2 through 12, selects sum 8, and projects 200 independent rolls.
General formula: N = s^dw(k) = sum[j=0..floor((k-d)/s)] (-1)^j C(d,j)C(k-sj-1,d-1)p(k) = w(k)/NH = np(k)Delta = 100p(k)-B There are s choices for each of d independent dice. Inclusion-exclusion counts ordered combinations for exact sum k, and division by the complete outcome count produces its probability.
What each symbol means
Worked substitution with the default inputs
With two fair six-sided dice, sum 8 has 5 of 36 outcomes, a 13.8889% probability, and 27.78 expected hits in 200 rolls.
Purpose-built visual
Explore a discrete dice-sum histogram
Every sum has a separate probability bar, with the selected total highlighted and the comparison threshold drawn across the distribution.
Worked situations
Practical examples
- Two d6 dice have 36 ordered outcomes.
- Exactly five combinations total 8.
- Middle sums have more ways than edge sums.
Better inputs
Useful tips
- Display the complete possible interval when auditing probabilities.
- Use exact probability for one sum.
- Do not round probabilities before calculating expected hits.
Before relying on the result
Limitations and common mistakes
- Dice are independent, fair, and identical.
- Rerolls and conditional rules are excluded.
- Expected frequency is not a prediction for one short session.
Reference
Key terms
- Outcome space
- All possible ordered results.
- Favorable ways
- Outcomes satisfying the selected sum.
- Discrete distribution
- Probability assigned to separate countable totals.
- Expected hits
- Trials multiplied by event probability.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Are all sums equally likely?
No. Middle totals can be formed in more ways.
Why count ordered outcomes?
Each die can show a different face, so (2,6) and (6,2) are separate outcomes.
Will 200 rolls produce exactly 27.78 hits?
No. That is the long-run expected count.
Why use bars instead of a line?
Dice sums are discrete outcomes, not a continuous measurement.