Probability · Dice & games
Dice Probability Calculator
The whole probability distribution of a dice total — not just the average. Set any number of dice and sides (the “NdS” of 3d6 or 13d8), add a modifier, and see the expected value, the spread, the most likely roll, and the exact chance of hitting a target. Plus advantage and disadvantage.
Roll several dice and keep the best (advantage) or worst (disadvantage) of them — e.g. D&D advantage is 2d20 keep highest 1.
Result
In plain English
Rolling dice doesn't produce “an average” — it produces a distribution, a whole spread of possible totals, each with its own chance. The average is just the balance point of that spread. The more dice you add, the more the total piles up in the middle (and the rarer the extremes), which is the central limit theorem happening in your hands.
- average (expected value)
- The long-run mean total: for N dice of S sides it is N·(S+1)/2, plus your modifier. It's the balance point — not the roll you should “expect” on any given throw.
- standard deviation
- How much a typical roll strays from the average. Bigger = a wider, flatter spread of outcomes.
- most likely total
- The single highest bar — the mode. For several dice it sits at (or next to) the average; for one die every face is equally likely.
- at least / at most / exactly
- The probability the total clears, stays under, or lands exactly on your target — the shaded area of the distribution.
- advantage / disadvantage
- Roll several dice and keep the best (or worst). Keeping the best shifts the odds up — but by less than beginners expect, and never past the single-die maximum.
- independence
- Dice have no memory. A run of low rolls does not make a high roll “due” — every throw starts the distribution over.
Frequently asked
What's the average of NdS dice (e.g. 13d8)?
N·(S+1)/2, plus any modifier. Each die of S sides averages (S+1)/2, so 13d8 averages 13 × 4.5 = 58.5. But the average is the balance point of the whole distribution — most single rolls land somewhere either side of it, never on a fraction.
If I just rolled three low numbers, is a high roll “due”?
No — that's the gambler's fallacy. Fair dice are independent and have no memory; the next roll faces the exact same distribution as the first. A streak feels like it must “balance out,” but the dice aren't keeping score.
How much does advantage (roll two, keep the highest) help?
On a d20 it raises the average from 10.5 to about 13.8 — worth roughly a +3 to +4 on a middling target, but less near the extremes, and it can never beat a natural 20. Use the Advantage tab to see the exact shift for your dice.
Why is rolling a 7 on two dice more likely than rolling a 12?
Because there are more ways to make it. A 12 needs both dice to show 6 — one combination out of 36. A 7 can come from 1-6, 2-5, 3-4, 4-3, 5-2 or 6-1 — six combinations, so six times as likely. Totals in the middle have the most combinations, which is exactly why the distribution of a dice sum peaks in the centre and tapers at the extremes.