Probability
Dice Sum Odds Calculator
Count exact ways to roll a target sum with identical fair dice, calculate probability and odds, display the full sum distribution, and extend the entered payoff assumptions across planned rolls.
Ways to reach exact target-
Total equally likely outcomes-
Exact target-sum probability-
Exact probability as percentage-
Expected successful rolls-
Expected net result across planned rolls-
Probability minus entered comparison-
Decision view
Exact dice-sum probability distribution
Exact dice-sum probability distributionEvery attainable sum is shown, with the entered target highlighted and the comparison probability marked.
Exact scenario comparisonExact target sum changes while all other entered assumptions remain constant.
| Exact target sum | Ways to reach exact target | Total equally likely outcomes | Exact target-sum probability | Exact probability as percentage | Expected successful rolls | Expected net result across planned rolls | Probability minus entered comparison |
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How to use Dice Sum Odds Calculator
- Enter the number of identical dice, sides per die, and exact target sum.
- Enter planned independent rolls and the payoff assumptions.
- Add a comparison probability if useful.
- Read the full probability histogram instead of judging the target in isolation.
Calculator guide
Understanding Dice Sum Odds Calculator
Exact dice-sum probability comes from counting equally likely ordered outcomes, not from assuming every possible total is equally likely. The complete distribution shows why center sums occur more often than edge sums and supports a transparent expected-payoff calculation.
Calculation method
How the calculation works
Count exact sum combinations for identical fair dice, divide by the complete equally likely outcome space, and extend the probability to odds, expected hits, and entered payoff arithmetic. Count ordered outcomes that equal the target, divide by sⁿ total outcomes, convert to odds, and multiply success and failure counts by their entered net payoff and stake.
Detailed calculation process
Count exact dice-sum outcomes before extending to payoff
The default rolls two fair six-sided dice, targets a total of 8, plans 100 independent rolls, risks $5 on failure, and receives a $20 net win on success.
General formula: Ω = sⁿF(tn,s) = Σ(-1)^k C(n,k)C(t-sk-1,n-1)p = F/ΩO_against = (1-p)/pE[X] = NpEV = N[pW-(1-p)S] The bounded-composition count F gives ordered ways for n identical-sided dice to sum to t. Dividing by the complete outcome space gives exact probability. Expected value uses the entered net win on success and stake loss on failure.
What each symbol means
Worked substitution with the default inputs
For two fair six-sided dice, sum 8 has 5 of 36 ordered outcomes, or 13.8889%; the entered payoff assumptions produce an expected net result of -$152.78 over 100 rolls.
Exact distribution
See every attainable dice sum and its probability
A probability histogram displays the complete combinatorial distribution, highlights the entered target, and marks the comparison probability.
Worked situations
Practical examples
- Two six-sided dice have 36 ordered outcomes.
- Five ordered pairs sum to 8.
- At the entered payoff, expected failure losses exceed expected success winnings.
Better inputs
Useful tips
- Confirm whether the game pays net profit or total return.
- Use the full distribution when comparing several target sums.
- Do not treat expected value as a guarantee over a short sequence.
Before relying on the result
Limitations and common mistakes
- The calculation assumes fair, independent dice with identical side counts.
- Real game rules can change payout, stake loss, pushes, bonuses, limits, and eligible outcomes.
- Expected value does not describe bankroll volatility, loss streaks, or responsible-play limits.
Reference
Key terms
- Ordered outcome
- A face result for each die where die positions are distinguishable.
- Favorable ways
- Ordered outcomes whose faces sum to the target.
- Odds against
- Unfavorable probability divided by favorable probability.
- Expected value
- Probability-weighted average net result over repeated trials.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why are dice sums not equally likely?
Different totals can be formed by different numbers of ordered face combinations.
Why is the expected success count fractional?
It is a probability-weighted average across repeated samples, not a predicted literal count.
What happens when the target is impossible?
Favorable ways and probability become zero, so odds against are not meaningfully finite.
Does a negative expected value predict my exact loss?
No. It describes the long-run average under the entered assumptions.