Dice Probability Calculator
Exact odds of rolling a total with any number of dice and any number of sides — exactly, at least or at most — with the full distribution table and chart.
Dice Probability Calculator: with the default inputs, probability is 16.666667%.
6 for standard dice; 20 for a d20.
- As a decimal
- 0.16666667
- Odds
- 1 in 6
- Exact fraction
- 6/36 = 1/6
- Average total
- 7
- Standard deviation of the total
- 2.4152
Assumptions
- Dice are fair and independent, and every face is equally likely.
- Only the sum of the dice is modelled — no rerolls, exploding dice or drop-lowest rules.
- Exact integer counts are produced for up to 8 dice with up to 20 sides; larger pools are computed as probabilities in double precision.
| Total | Ways to roll it | Probability | Cumulative |
|---|---|---|---|
| 2 | 1 | 2.7778% | 2.7778% |
| 3 | 2 | 5.5556% | 8.3333% |
| 4 | 3 | 8.3333% | 16.6667% |
| 5 | 4 | 11.1111% | 27.7778% |
| 6 | 5 | 13.8889% | 41.6667% |
| 7 | 6 | 16.6667% | 58.3333% |
| 8 | 5 | 13.8889% | 72.2222% |
| 9 | 4 | 11.1111% | 83.3333% |
| 10 | 3 | 8.3333% | 91.6667% |
| 11 | 2 | 5.5556% | 97.2222% |
| 12 | 1 | 2.7778% | 100% |
Totals in the middle have far more ways of happening than the extremes — that is why 7 is the most likely roll of two dice and 2 the least.
How this is worked out
The formula
Total outcomes with n dice of s sides = sⁿ (every ordered roll is equally likely). The number of ways to reach each total is the convolution of n copies of the uniform distribution on 1…s — the same "add the previous row" idea as Pascal's triangle, and what the distribution table below lists. Mean total = n(s + 1) ÷ 2 Standard deviation = √( n(s² − 1) ÷ 12 )
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Number of dice
- A number.from 1 to 20 · whole numbers only · defaults to 2
- Sides per die
- 6 for standard dice; 20 for a d20.from 2 to 120 · whole numbers only · defaults to 6
- Target total
- A number.from 1 to 2400 · whole numbers only · defaults to 7
- You want a total that is
- Choose one of 3 options.Exactly the target · At least the target · At most the target
What you get back
- Probabilitymain answer
- As a decimal
- Odds
- Exact fraction
- Average total
- Standard deviation of the total
What this assumes
- Dice are fair and independent, and every face is equally likely.
- Only the sum of the dice is modelled — no rerolls, exploding dice or drop-lowest rules.
- Exact integer counts are produced for up to 8 dice with up to 20 sides; larger pools are computed as probabilities in double precision.
About this calculator
Dice totals are not uniform, and that is the whole reason board games use them. There is one way to roll a 2 with two dice and six ways to roll a 7, so a 7 comes up six times as often. This calculator gives the exact odds for any number of dice with any number of sides, and shows the full shape of the distribution behind the answer.
How to use it
Set the number of dice, the sides per die, and the total you care about, then choose whether you need exactly that total, at least it, or at most it. The "at least" mode is the one you want for the standard tabletop question, "do I beat a DC of 15?"
How the odds are worked out
Every ordered roll of n dice with s sides is equally likely, and there are sⁿ of them. Counting how many produce each total is a convolution: start with one die's flat distribution, then repeatedly combine it with the next die's. The table lists those counts directly, so for two six-sided dice you can see the familiar 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 running from 2 to 12.
Up to eight dice with twenty or fewer sides, the counts are exact integers computed with arbitrary-precision arithmetic, so the fraction shown really is exact — 6/36 = 1/6 for a 7 on two dice, with no rounding anywhere.
Reading the distribution
- With one die every total is equally likely — a flat line.
- With two the shape is a triangle, peaking at the middle total.
- With three or more it starts to look like a bell curve. That is the central limit theorem arriving early: by five dice the normal approximation is already good to a couple of percent, which is why the calculator caps at twenty and points you at a z-score beyond that.
The mean is always n(s + 1)/2 — 7 for two d6 — and the spread grows only as √n, so more dice means a relatively tighter result. Rolling 4d6 for a total of 14 is far more predictable than rolling 1d24.
Common mistakes
- "Two dice have 11 outcomes, so each is 1 in 11." No: there are 36 ordered outcomes, unevenly distributed across the 11 totals.
- Confusing "at least one 6" with "a total of at least 6". This calculator answers the second. For the first, use the probability calculator's at-least-one mode: 1 − (5/6)ⁿ.
- Assuming dice are due. Each roll is independent; a run of low totals changes nothing about the next one.
- Forgetting that dice pools with rerolls, exploding dice or "drop the lowest" rules are different distributions entirely — those need a simulation, not this formula.
Frequently asked questions
▸What are the odds of rolling a 7 with two dice?
Exactly 6 in 36, which is 1 in 6 or 16.67%. Seven has more combinations than any other total, which is why it is the most common roll.
▸Why isn't every dice total equally likely?
Because totals in the middle can be made in more ways. There is one way to make 2 (1+1) but six ways to make 7 (1+6, 2+5, 3+4 and their reverses).
▸What are the odds of rolling 18 on 3d6?
1 in 216 — a single outcome out of 6³. The same odds apply to rolling 3.
▸How do I work out 'at least' odds?
Add the probabilities of every total from your target upwards. The calculator does that directly, and the cumulative column in the table shows the running totals.
▸Does rolling more dice make the result more predictable?
Relatively, yes. The average grows in proportion to the number of dice but the spread only grows as its square root, so the total clusters more tightly around the mean as a percentage.
Put this calculator on your own site
A working dice odds, free for any site, with no ads and no sign-up. It resizes to fit wherever you paste it and updates itself as this page improves.
Paste this anywhere. It works on any site, carries no ads, never expires, and always shows the current version.
Dice Probability Calculator by CalculateItNow
The page's own title. The clearest description of what the link leads to.
The credit line sits outside the widget on purpose, so it is a real link on your page rather than one buried in a frame. Please keep it — it is what pays for CalculateItNow staying free and ad-free. The script only resizes the widget to fit its contents; drop it and the widget still works.
Browse every calculator widget·How to add it to WordPress, Squarespace or Wix
Related calculators
The questions people ask next to a dice odds.
Single-event, 'A and B', 'A or B', at-least-one and binomial (exactly k of n) probabilities as percentages, fractions and odds, with the rule applied shown.
Exact jackpot odds for any pick-n-from-m lottery, with or without a bonus ball, plus the odds of every smaller prize tier — computed with exact combinatorics.
nCr and nPr, with or without repetition — exact answers for huge counts, the formula substituted, and a plain explanation of when order matters.
Draw whole numbers from any range, with or without repeats, from an explicit seed — so the same seed always reproduces the same draw and anyone can check it.