Inputs
Tip
Use Add Group to model sets like 2d6 + 1d8. Advanced rules are simulated; the simple case is exact.
| # | Dice (N) | Sides (S) | Reroll Ones | Explode on Max | Adv./Dis. | Actions |
|---|
Added after all dice.
Drop this many lowest dice total.
Used when advanced rules are active.
Results
| Total dice (all groups) | 0 |
|---|---|
| Exact expected value | N/A |
| Simulated mean | — |
| Simulated standard deviation | — |
| 95% CI for mean | — |
| Notes | Exact shown only when no advanced rules are active. |
Example Data
| Scenario | Groups (N×S) | Modifier | Specials | Estimated Average |
|---|---|---|---|---|
| Classic two dice | 2×6 | +0 | None | 7.0 |
| Three d8 damage | 3×8 | +0 | None | 13.5 |
| Single d20 with advantage | 1×20 | +0 | Advantage | ≈13.83 |
Formulas Used
Simple case (exact): For a fair die with S sides, the expected value of one roll is (S + 1) / 2. For a group with N dice, expectation is N × (S + 1) / 2. With multiple groups, sum the expectations and then add any modifier:
E[total] = Σgroups i Ni · (Si + 1)/2 + modifier
Advanced rules (simulated): When any of the following are active, results are estimated via Monte Carlo simulation: reroll ones, exploding on max faces, advantage/disadvantage, or dropping lowest dice. The simulation rolls many trials and reports the sample mean and a 95% confidence interval.
Assumptions: all dice are discrete uniform 1..S and independent; advanced options interact as documented in the notes below.
How to Use This Calculator
- Start with one or more groups, e.g., 2d6 and 1d8.
- Optionally set Reroll Ones, Explode on Max, or Adv./Dis. per group.
- Set any overall Modifier (added after dice) and Drop Lowest count.
- Choose a number of Trials for the simulation when advanced rules apply.
- Click Compute. The exact expectation appears when no advanced rules are active.
- Export your results using Download CSV or Download PDF.
Interaction rules: If Adv./Dis. is set for a group, exploding and reroll options for that group are ignored during simulation (they rarely coexist in common systems).
FAQs
For an S‑sided fair die, the average outcome is (S + 1) / 2.
Only when all groups use no advanced rules and Drop Lowest is zero. Otherwise, the calculator switches to Monte Carlo simulation.
Accuracy improves with more trials. We show a 95% confidence interval to indicate uncertainty in the estimated mean.
If a die rolls its maximum face, you add the maximum and roll again, repeating while maximums continue. This raises the expected total.
You may choose Once (reroll a 1 one time) or Until Success (keep rerolling while a 1 appears). We apply this per individual die roll.
They roll two dice of the same type: advantage takes the higher result; disadvantage takes the lower result. This calculator simulates that behavior for any S‑sided die.
We simulate dropping the specified number of lowest individual dice across all groups before adding the modifier. Exact formulas are complex, so we estimate via trials.
Reference: Common Expected Values
Exact expectations for typical fair dice; no modifiers.
| Sides (S) | E[1dS] | E[2dS] | E[3dS] |
|---|---|---|---|
| 4 | 2.5 | 5 | 7.5 |
| 6 | 3.5 | 7 | 10.5 |
| 8 | 4.5 | 9 | 13.5 |
| 10 | 5.5 | 11 | 16.5 |
| 12 | 6.5 | 13 | 19.5 |
| 20 | 10.5 | 21 | 31.5 |
Reference: Advantage, Disadvantage, Exploding
Single die expectations by rule variants.
| Sides (S) | E[1dS] | E[Advantage] | E[Disadvantage] | E[Exploding on Max] |
|---|---|---|---|---|
| 4 | 2.5 | 3.125 | 1.875 | 3.333 |
| 6 | 3.5 | 4.472 | 2.528 | 4.2 |
| 8 | 4.5 | 5.812 | 3.188 | 5.143 |
| 10 | 5.5 | 7.15 | 3.85 | 6.111 |
| 12 | 6.5 | 8.486 | 4.514 | 7.091 |
| 20 | 10.5 | 13.825 | 7.175 | 11.053 |