Understanding the Dice Roller
This tool rolls virtual dice using standard tabletop notation and reports both a random result and the exact statistics of that roll. Dice notation is written as NdS+M, where N is the number of dice, S is the number of sides on each die, and M is an optional flat modifier added to (or subtracted from) the sum. For example, 3d6+2 means "roll three six-sided dice, add them together, then add 2."
The formulas
For a roll of N dice with S sides each plus a modifier M, three key values are exact and do not require simulation:
- Minimum total: every die shows 1, so the lowest possible sum is N + M.
- Maximum total: every die shows its highest face S, so the peak is N × S + M.
- Average (expected value): a single fair S-sided die averages (S+1)/2, so N dice plus a modifier average N × (S+1)/2 + M.
The random result itself is generated by rolling each die independently and uniformly — every face from 1 to S is equally likely — then summing and applying the modifier. Because the dice are independent, the totals cluster around the average and form a bell-shaped distribution as N grows (a consequence of the Central Limit Theorem).
Common reference points
- 1d6 (one standard die): range 1–6, average 3.5.
- 2d6 (two dice, as in many board games): range 2–12, average 7 — 7 is the single most likely sum.
- 3d6 (classic ability-score roll): range 3–18, average 10.5.
- 1d20 (a twenty-sided die): range 1–20, average 10.5.
- 1d100 (percentile die): range 1–100, average 50.5.
Why the distribution matters
Rolling one die gives a flat (uniform) distribution: every outcome is equally likely. Rolling several dice and summing does not. With 2d6, a total of 7 can happen six ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) but a total of 2 only one way (1+1), so 7 is six times more likely than 2. Adding more dice narrows the spread relative to the average, which is why 3d6 produces middling scores far more often than extreme ones.