Choose coin flips with custom p(Heads), weighted six-sided dice, or European/American roulette bets. The simulator compares theory and empirical ratios, shows Wilson 95% intervals for binary outcomes, tracks convergence, and records each calculation step.
For handouts, export raw counts as CSV or copy the LaTeX summary. Use the shareable URL to reopen the same scenario later.
Controls
Distribution (theory vs empirical)
Blue bars track empirical frequencies while the lighter overlay shows theoretical probabilities. Differences are labeled above each category for quick reference.
Running estimate vs trials
The solid line shows the running proportion. The dashed line marks the theoretical value and the shaded band gives the 95% confidence interval as trials accumulate.
Teaching tips
- Demonstrate the law of large numbers by increasing trials and letting learners watch the convergence band tighten around the theoretical line.
- Switch the dice to weighted mode and discuss how expected frequencies change and how the χ² statistic reacts to mis-specified weights.
- Highlight roulette's expected value and house edge to connect probability with finance-style expected returns.
How to use this calculator effectively
This guide helps you use Probability Simulator — coin, dice, roulette (theory vs empirical) in a repeatable way: define a baseline, change one variable at a time, and interpret outputs with explicit assumptions before you share or act on results.
How it works
The page applies deterministic logic to your inputs and shows rounded output for readability. Treat it as a comparison workflow: run one baseline case, adjust a single parameter, and measure both absolute and percentage deltas. If a result seems off, verify units, time basis, and sign conventions before drawing conclusions. This approach keeps your analysis reproducible across teammates and sessions.
When to use
Use this page when you need a fast estimate, a classroom check, or a practical what-if comparison. It works best for planning and prioritization steps where you need direction and magnitude quickly before investing in deeper modeling, manual spreadsheets, or formal external review.
Common mistakes to avoid
- Changing multiple parameters at once, which hides the true cause of output movement.
- Mixing units (percent vs decimal, monthly vs yearly, gross vs net) across scenarios.
- Comparing with another tool without aligning defaults, constants, and rounding rules.
- Using rounded display values as exact downstream inputs without re-checking precision.
Interpretation and worked example
Run a baseline scenario and keep that result visible. Next, modify one assumption to reflect your realistic alternative and compare direction plus size of change. If the direction matches your domain expectation and the size is plausible, your setup is usually coherent. If not, check hidden defaults, boundary conditions, and interpretation notes before deciding which scenario to adopt.
See also
FAQ
How can I verify the RNG sequence in class?
Use a published acceptance case, such as seed 12345 with 100 flips. The simulator returns the same outcome each time because RNG and seed are fixed, even across devices.
Can I export the run to share with students?
Yes. Export CSV for raw counts and running estimates, copy the LaTeX summary for worksheets, and share the generated URL to reload the same experiment.
What should I do first on this page?
Start with the minimum required inputs or the first action shown near the primary button. Keep optional settings at defaults for a baseline run, then change one setting at a time so you can explain what caused each output change.
Why does this page differ from another tool?
Different pages often use different defaults, units, rounding rules, or assumptions. Align those settings before comparing outputs. If differences remain, compare each intermediate step rather than only the final number.
How reliable are the displayed values?
Values are computed in the browser and rounded for display. They are good for planning and educational checks, but for regulated or high-stakes decisions you should validate assumptions with official guidance or professional review.