Probability
Dice Outcome Confidence Interval Calculator
Calculate observed dice-outcome proportion, design-adjusted standard error, margin, clipped interval bounds, theoretical gap, future expected count, and a visual confidence interval.
Observed success proportion-
Design-adjusted standard error-
Critical-value margin-
Lower probability bound clipped at zero-
Upper probability bound clipped at one-
Entered theoretical probability as decimal-
Observed minus theoretical probability-
Future successes at observed rate-
Margin expressed in percentage points-
Calculated margin minus entered comparison-
Decision view
Observed dice probability confidence interval
Observed dice probability confidence intervalLower and upper bounds, observed rate, theoretical benchmark, and margin comparison share one probability view.
Exact scenario comparisonObserved independent rolls changes while all other entered assumptions remain constant.
| Observed independent rolls | Observed success proportion | Design-adjusted standard error | Critical-value margin | Lower probability bound clipped at zero | Upper probability bound clipped at one | Entered theoretical probability as decimal | Observed minus theoretical probability | Future successes at observed rate | Margin expressed in percentage points | Calculated margin minus entered comparison |
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How to use Dice Outcome Confidence Interval Calculator
- Enter observed target outcomes and the number of independent rolls.
- Enter the critical value and any justified overdispersion factor.
- Enter a theoretical benchmark, future roll count, and margin comparison.
- Read the interval, observed point, and theoretical marker on the probability axis.
Calculator guide
Understanding Dice Outcome Confidence Interval Calculator
An observed dice-outcome rate is a sample proportion, while a theoretical probability is a model benchmark. This page keeps them separate and shows a transparent normal-approximation interval with an entered critical value and design-effect multiplier.
Calculation method
How the calculation works
Calculate a transparent normal-approximation interval for an observed dice outcome proportion, keep theoretical probability separate, and extend only the observed rate to a future count. Divide observed successes by rolls, calculate the Bernoulli standard error with the entered design effect, multiply by the critical value, clip bounds to zero and one, and compare theory separately.
Detailed calculation process
Build a design-adjusted interval around an observed dice rate
The default observes 34 target outcomes in 200 rolls, uses z=1.96 and design effect 1, compares with 16.667%, and extends the observed rate to 500 future rolls.
General formula: p̂ = x/nSE = √[p̂(1-p̂)D/n]m = zSEL = max(p̂-m,0)U = min(p̂+m,1)Δ = p̂-p₀E_future = N_fp̂ The observed proportion supplies the center. Bernoulli variability, sample size, and the entered overdispersion factor determine standard error. The critical value scales that error into a margin; clipping prevents impossible probability bounds.
What each symbol means
Worked substitution with the default inputs
The default observed rate is 17.00% with an approximate 11.794% to 22.206% interval; the 16.667% theoretical benchmark lies inside that interval.
Uncertainty interval
Place the observed rate, interval, and theory on one axis
A confidence-interval dot plot shows the lower and upper bounds, the observed center, the theoretical marker, and the entered margin threshold without converting them into unrelated cards.
Worked situations
Practical examples
- Thirty-four outcomes in 200 rolls give 17.0%.
- The default standard error is 2.656 percentage points.
- The normal interval spans about 11.794% to 22.206%.
Better inputs
Useful tips
- Define the target event before examining the data.
- Use an interval method appropriate to sample size and purpose.
- Treat a design effect above one only as a documented dependence adjustment.
Before relying on the result
Limitations and common mistakes
- The displayed interval is a plug-in normal approximation and can perform poorly with small counts or probabilities near zero or one.
- Dependence, optional stopping, changing dice, multiple comparisons, and post-selected outcomes invalidate the simple model.
- A theoretical probability inside the interval is not proof that the dice are fair.
Reference
Key terms
- Sample proportion
- Observed target count divided by observed rolls.
- Standard error
- Modeled sampling variability of the observed proportion.
- Critical value
- Multiplier converting standard error into the selected margin convention.
- Design effect
- Entered multiplier for variance relative to independent Bernoulli trials.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Does 95% mean a 95% chance the true probability is inside this one interval?
Under frequentist interpretation, the procedure has long-run coverage under its assumptions; it is not a posterior probability statement.
Why clip the bounds?
Probabilities cannot be below 0 or above 1.
Why use design effect?
It can transparently inflate variance when observations are more variable than independent Bernoulli trials, but the value must be justified.
Should I use an exact interval?
For small or extreme samples, an exact or other binomial interval may be more appropriate.