Probability
Dice Simulation Calculator
Calculate exact outcome counts, expected hits, entered observed rate, hit difference, and an approximate standard error for a reproducible dice experiment.
Exact equally likely outcomes-
Exact ways to reach target-
Exact target probability-
Expected target hits in planned trials-
Observed hit rate-
Observed minus expected hits-
Approximate probability standard error-
Decision view
Exact dice distribution versus observed run
Exact dice distribution versus observed runThe exact target probability and entered simulation rate share one probability scale with an explicit standard-error band.
Exact scenario comparisonPlanned simulated trials changes while all other entered assumptions remain constant.
| Planned simulated trials | Exact equally likely outcomes | Exact ways to reach target | Exact target probability | Expected target hits in planned trials | Observed hit rate | Observed minus expected hits | Approximate probability standard error |
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How to use Dice Simulation Calculator
- Specify independent fair dice and a reachable target.
- Run the stated number of trials externally and enter observed hits.
- Interpret deviations relative to standard error, not by raw hit count alone.
Calculator guide
Understanding Dice Simulation Calculator
A simulation result should be compared with the exact fair-dice probability and the sampling variation expected from its trial count.
Calculation method
How the calculation works
Compare an exact fair-dice probability with the expected and entered observed result from a simulation run, including a sampling-error reference. Count exact ways to reach the target, divide by all equally likely outcomes, multiply by planned trials for expected hits, and compare the entered run with binomial standard error.
Detailed calculation process
Compare the entered simulation run with the exact dice model
The default evaluates 10,000 trials of two fair six-sided dice for a target total of seven, with 1,650 entered target hits.
General formula: N = s^d; F = ways(d,s,t); p = F/N; E = np; p_obs = min(h,n)/n; delta = min(h,n)-E; SE = sqrt(p(1-p)/n) Combinatorics supplies the exact probability independently of the entered simulation. Expected hits and sampling error then provide a scale for interpreting the observed run.
What each symbol means
Worked substitution with the default inputs
The default observed result is close to the exact fair-dice benchmark; the difference is small relative to the expected sampling variation.
Simulation audit
Make the experiment reproducible
A credible comparison includes more than the final hit count.
Worked situations
Practical examples
- Two six-sided dice have 36 equally likely ordered outcomes.
- A target of seven has six favorable outcomes.
- Doubling trials reduces standard error by about the square root of two.
Better inputs
Useful tips
- Record the random seed.
- Use many trials for rare totals.
- Check the generator and counting code when deviations persist.
Before relying on the result
Limitations and common mistakes
- The calculator does not generate trials.
- The normal standard-error approximation can be weak for rare events or small samples.
- Dice are assumed independent and fair.
Reference
Key terms
- Exact probability
- Favorable equally likely outcomes divided by all outcomes.
- Expected hits
- Trials multiplied by exact probability.
- Observed rate
- Entered hits divided by trials.
- Standard error
- Approximate run-to-run variation of a sample proportion.
Important note
The exact benchmark assumes independent fair dice with the entered side count. The observed result is supplied by the user, and the normal standard-error approximation can be weak for rare targets or small runs.
Frequently asked questions
Does the calculator roll dice?
No. It evaluates an entered simulation run.
Why use ordered outcomes?
Each die is distinct in the exact sample space.
Does a deviation prove bias?
No. Sampling variation must be considered.
What happens with an impossible target?
The exact favorable count is zero.