What does this simulator compute?

It computes the probability of every possible total when rolling n dice of s sides, drawing the exact theoretical distribution, an empirical histogram from simulated rolls, and a matching Gaussian curve so you can compare theory with experiment.

How is the exact distribution calculated?

The single-die uniform distribution is convolved with itself once per extra die, counting face-combination totals, then divided by s to the power n to give exact probabilities with no approximation.

Why does the histogram become bell-shaped?

This is the central limit theorem: summing many independent identically distributed variables produces a distribution that tends towards a normal curve, even though a single die is uniform.

Home▸Board Games & Game Mathematics▸Dice & Probability (2D)

🎲 Dice & Probability (2D)

2D Canvas dice-sum probability lab: roll n dice with s sides, compare the exact convolution PMF against a live empirical histogram and Gaussian overlay, and watch casino bet expected values update in real time.

Board Games & Game Mathematics2DEasy60 FPS📱 Mobile-adapted⇄ 3D version

This 2D companion drives the same exact-convolution dice-sum maths as the 3D version through a plain Canvas view built for reading the statistics rather than watching a WebGL scene: a side panel sets the number of dice, sides per die, and auto-roll speed, while the canvas stack shows the currently rolled faces above a live histogram comparing the empirical distribution against the exact theoretical PMF and an optional Gaussian overlay, with a casino-odds table for two six-sided dice updating alongside the live mean, variance and mode readouts.

⚙ Under the hood

2D dice-sum probability lab: exact n-fold convolution PMF, live empirical histogram, Gaussian overlay and casino bet expected values, all on Canvas 2D.

dice probabilitysum of dice distributioncentral limit theoremlaw of large numbersexpected valuecasino probability

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install