Probability Simulator (Coin / Dice / Cards)

Why simulate probability?

A probability simulator is a practical way to understand randomness. Many problems have clean mathematical formulas, but the intuition behind them can be hard to build from symbols alone. By simulating coin flips, dice rolls, or card draws thousands of times, you can see patterns emerge and develop a stronger sense of what outcomes are common, what outcomes are rare, and how probability behaves as you repeat an experiment.

Theoretical probability vs simulated probability

When an experiment has a known distribution, you can compute theoretical probabilities. For example, the number of heads in n fair coin flips follows a binomial distribution. The number of hearts in a hand of cards without replacement follows a hypergeometric distribution. This tool overlays the theoretical distribution with the simulated histogram so you can compare them directly.

Law of large numbers and convergence

One of the most important ideas in probability is the law of large numbers: as the number of trials increases, the observed frequency of an outcome tends to get closer to the true probability. That does not mean the results become perfectly smooth or predictable; it means the average behavior becomes more stable over time. The convergence chart in this simulator shows a running estimate for a chosen outcome so you can watch it settle.

Classroom and practical uses

Students can use this probability simulator to verify homework answers, explore “what if” questions, and understand distributions like sums of dice rolls. In practice, simulation is also used in finance, operations research, engineering, and risk modeling, especially when a closed-form solution is hard or when you want to validate assumptions with a model.

This tool runs entirely in the browser and does not upload your data to a server. It saves your last-used scenario and settings in localStorage for convenience (clearing your browser data removes that history).