Probability
Dice Risk Calculator
Calculate exact adverse dice-sum probability, repeated-trial occurrence risk, expected loss, residual loss after mitigation, and the expected economic value of the mitigation plan.
Equally likely outcomes per trial-
Outcomes in the adverse tail-
Adverse probability per trial-
Expected adverse outcomes-
Chance of at least one adverse outcome-
Expected loss before mitigation-
Residual loss per adverse outcome-
Mitigated expected loss plus fixed cost-
Expected value of mitigation-
Decision view
Exact dice-sum risk tail
Exact dice-sum risk tailEvery possible sum is plotted from its exact combination count; adverse bars, the entered threshold, repeated-trial chance, and mitigation economics remain linked to the current inputs.
Exact scenario comparisonAdverse outcome: total at or below changes while all other entered assumptions remain constant.
| Adverse outcome: total at or below | Equally likely outcomes per trial | Outcomes in the adverse tail | Adverse probability per trial | Expected adverse outcomes | Chance of at least one adverse outcome | Expected loss before mitigation | Residual loss per adverse outcome | Mitigated expected loss plus fixed cost | Expected value of mitigation |
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How to use Dice Risk Calculator
- Choose the number and sides of the dice.
- Define the adverse total and number of repeated trials.
- Enter consequence and mitigation assumptions, then compare the two expected-cost paths.
Calculator guide
Understanding Dice Risk Calculator
Dice risk is discrete: changing the adverse-sum cutoff by one point adds a whole layer of combinations. This page enumerates that exact lower tail before extending it to repeated trials and loss mitigation.
Detailed calculation process
Detailed exact dice-risk calculation
The default case uses two fair six-sided dice, an adverse total of four or less, and 100 independent trials.
General formula: W=s^dA=sum(c_x, x<=t)p=A/WE=npP(>=1)=1-(1-p)^nL_0=E*lL_1=E*l(1-m)+C All face configurations are equally likely. Exact adverse combinations produce the per-trial risk, which then drives repeated-trial probability and both loss paths.
What each symbol means
Worked substitution with the default inputs
The six adverse combinations reconcile to 16.667% per trial, and the expected loss difference is $1,250 - $1,650 = -$400.
Worked situations
Practical examples
- For two six-sided dice, totals of 2, 3, or 4 occupy 6 of 36 combinations, so the per-roll adverse probability is 16.667%.
- Across 100 independent trials, even a modest per-trial tail makes at least one adverse result overwhelmingly likely.
Better inputs
Useful tips
- Define the adverse event before inspecting outcomes.
- Use the exact dice configuration rather than a normal approximation for small dice pools.
- Treat mitigation effectiveness as a reduction in consequence, not probability, unless the control actually changes the roll mechanism.
Before relying on the result
Limitations and common mistakes
- The repeated-trial formula assumes independent identically distributed rolls.
- Loaded dice, reroll rules, conditional play, and changing stakes require a different event model.
- Expected loss does not show the full distribution of realized loss.
Reference
Key terms
- Combination count
- Number of equally likely face configurations producing a total.
- Adverse tail
- All totals at or below the entered threshold.
- Complement rule
- At-least-one probability calculated as one minus the zero-event probability.
Important note
Use the same definition of an adverse outcome for the probability and consequence assumptions.
Frequently asked questions
Why is the risk curve stepped?
Dice totals are integers, so a threshold change includes or excludes complete probability masses.
Does expected count predict the actual count?
No. It is the long-run average across repeated comparable plans.
When is mitigation economically favorable?
When expected unmitigated loss exceeds residual expected loss plus the fixed control cost.