Dice Simulator

On a fair n-sided die, every face has an equal probability of 1/n. On a d6 that is 1/6 ≈ 16.7%. On a d20 it is 1/20 = 5%. This assumes the die is perfectly balanced — each face is equally likely, and the die has no memory of previous rolls. The rolls are independent events.

The expected value (long-run average) of a single n-sided die is (n+1)/2. For a d6 this is 3.5; for a d20 it is 10.5. When rolling k dice, expected total = k × (n+1)/2. Rolling 3d6 averages 10.5, while rolling 2d20 averages 21. The simulator shows this for your current configuration.

This is the Central Limit Theorem: the sum of many independent random variables tends toward a bell-curve distribution, regardless of the individual distribution. With more dice, extreme results (all 1s or all max) become exponentially rarer, so most totals cluster near the expected value. Rolling 10d6 almost never gives a total below 15 or above 45, even though 6–60 is theoretically possible.

For one n-sided die, σ = √((n²−1)/12). For a d6 this is ≈ 1.71. When rolling k dice, the total standard deviation is √(k × (n²−1)/12). It grows with √k, not k: doubling the dice only makes the spread √2 ≈ 1.41× wider while the average doubles. This is why large pools of dice produce very consistent results relative to their range.

Each die covers a different outcome range. d4: 1–4, avg 2.5 — used for small damage. d6: 1–6, avg 3.5 — the everyday die, used in board games and many RPGs. d8: 1–8, avg 4.5. d10: 1–10 (or 0–9), avg 5.5. d12: 1–12, avg 6.5. d20: 1–20, avg 10.5 — the iconic RPG die for ability checks and attacks. d%: two d10 combined for 1–100, for percentage chances.