Dice Probability Calculator
Work out the odds of any roll: one die, the total of several dice, how often a number turns up over many rolls, and D&D and board game rolls like advantage, 4d6 drop the lowest and Yahtzee. Every answer shows the exact fraction and the working.
What do you want to know?
Examples: 3d6, 2d20+1, 1d8+1d6+2. Mix any dice and add or take away a fixed number.
Rolling one die many times and rolling many dice at once work the same way.
Results update as you type. Press Calculate or Enter to save one to your recent list.
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Practice questions
Five fresh questions each time, with worked answers. Teachers can print them as a worksheet with an answer key, or save it as a PDF from the print window.
Why two dice don't behave like one die
On one fair die every face is equally likely. Add a second die and that stops being true for the total. Only one roll gives 2 (a 1 and a 1), but six rolls give 7: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. So 7 comes up six times as often as 2, even though both dice are perfectly fair.
The more dice you add, the more the totals bunch up in the middle. With three dice, a 10 or an 11 each come up 1 time in 8, while a 3 or an 18 each come up just 1 time in 216.
Every total with two ordinary dice
| Total | Ways | Exact | Probability |
|---|---|---|---|
| 2 or 12 | 1 | 1/36 | 2.78% |
| 3 or 11 | 2 | 1/18 | 5.56% |
| 4 or 10 | 3 | 1/12 | 8.33% |
| 5 or 9 | 4 | 1/9 | 11.11% |
| 6 or 8 | 5 | 5/36 | 13.89% |
| 7 | 6 | 1/6 | 16.67% |
The formulas behind each mode
One die
P = faces that count / sides
"4 or higher" on a d6 counts the faces 4, 5 and 6, so the chance is 3/6, which is 1/2.
Dice total
P = ways to make the total / (sides₁ × sides₂ × ...)
Every combination of faces is equally likely. The calculator counts the ways to reach each total by adding one die at a time, which works for any mix of dice, such as 1d8 + 1d6 + 2.
Repeated rolls (binomial)
P(k hits in n rolls) = C(n, k) × p^k × (1 − p)^(n − k)
For "at least once", it's quicker to work out the chance of missing every time and subtract: 1 − (1 − p)^n.
Advantage and disadvantage
Advantage: 1 − ((t − 1)/s)² Disadvantage: ((s − t + 1)/s)²
With advantage you only fail if both dice miss. With disadvantage you only succeed if both dice hit.
Seeing every face
Average rolls = s × (1 + 1/2 + 1/3 + ... + 1/s)
For a d6 that's 14.7 rolls. Early faces come quickly, but the last one takes 6 rolls on average by itself.
Worked examples
A 7 with two dice
Two dice can land 6 × 6 = 36 ways, all equally likely. Six of them add up to 7, so the chance is 6/36, which is 1/6 or 16.67%. A total of 2 needs both dice to show 1, which is 1/36, about 2.78%.
At least one 6 in four rolls
It's tempting to say each roll gives a 1/6 chance, so four rolls give 4/6, about 66.7%. That's wrong, because it counts rolls with two or more sixes more than once. The chance of no 6 in four rolls is (5/6)^4, about 48.23%, so the chance of at least one 6 is about 51.77%.
This exact bet helped start probability theory. In 17th-century France, the Chevalier de Méré made money betting on at least one 6 in four rolls. He then bet on at least one double six in 24 rolls of two dice, reasoning it should be just as good. It isn't: that chance is only 49.14%. His question reached Blaise Pascal, whose letters with Pierre de Fermat in 1654 laid the groundwork for the mathematics of chance.
Advantage in D&D
Needing 15 or higher on a d20 is a 30% chance. With advantage it becomes 1 − (14/20)² = 51%. Advantage doesn't double your chance: a natural 20 goes from 5% to 9.75%, not 10%, because sometimes both dice show 20.
Rolling ability scores
Roll 4d6 and drop the lowest die, the classic way to roll D&D ability scores. The average score is 12.24, you'll get 15 or higher 23.15% of the time, and a perfect 18 turns up 7 times in 432, about 1.62%.
Common game odds
All figures assume fair six-sided dice unless stated.
| Situation | Exact | Probability | About 1 in |
|---|---|---|---|
| Doubles with two dice (Monopoly, backgammon) | 1/6 | 16.67% | 6 |
| A 6 or an 8 in Catan | 5/18 | 27.78% | 3.6 |
| 7 or 11 on the come-out roll in craps | 2/9 | 22.22% | 4.5 |
| Winning a pass line bet in craps | 244/495 | 49.29% | 2.03 |
| Yahtzee on the first roll | 1/1,296 | 0.0772% | 1,296 |
| Natural 20 on a d20 | 1/20 | 5% | 20 |
| Natural 20 with advantage | 39/400 | 9.75% | 10.3 |
| At least one 6 in 4 rolls | 671/1,296 | 51.77% | 1.93 |
| At least one double six in 24 rolls of two dice | 49.14% | 2.03 | |
| Every face of a d6 within 10 rolls | 27.18% | 3.68 |
How real dice behave
The maths on this page assumes perfectly fair dice. Real ones are close, but not always exact. In 1894, the biologist Walter Weldon rolled 12 dice 26,306 times, counting how many showed a 5 or a 6 each time, and sent the results to Francis Galton. Karl Pearson used this data in his 1900 paper introducing the chi-square test, and found that 5s and 6s turned up more often than fair dice would explain.
The likely reason is the pips. On ordinary dice they're scooped out, so the 6 face loses more material than the 1 opposite it and ends up slightly lighter, making it more likely to land face up. In 2009, Zacariah Labby repeated the experiment with a dice-rolling machine and didn't find the same effect. Casino dice avoid the problem altogether, with sharp edges and pips filled in with material of matching weight.
You can try Weldon's setup with the "5 or 6 on 4+ of 12 dice" example in Repeated rolls.
Mistakes that give the wrong answer
Treating every total as equally likely. That's only true for one die. With two dice, a 7 is six times as likely as a 2 or a 12.
Treating 2d6 like 1d12. Both go up to 12, but a d12 can roll a 1 and treats every number equally. Two d6 can't roll a 1, and a 7 is twice as likely as it would be on a d12.
Adding chances for "at least once". Four rolls don't give 4 × 1/6 = 66.7% for a 6. Subtract the chance of never hitting from 1 instead, which gives 51.77%.
Thinking a number is "due". Dice have no memory. After three 6s in a row, the next roll is still 1 in 6.
Assuming advantage doubles your chance. It helps most for middling targets and never quite doubles anything, as the natural 20 example shows.
How we calculate
Results use exact whole-number arithmetic (JavaScript BigInt) wherever the numbers allow, so the fraction you see is exact. Dice totals are counted by adding one die at a time. "Roll several, keep the highest" checks every possible way the dice can land. For very large numbers of rolls, the calculator switches to high-precision decimals and stops showing the fraction. Percentages are rounded only at the last step.
The simulation button rolls 10,000 sets of virtual dice in your browser and counts how often your target happens. It lands close to the exact answer but rarely on it, and the result tells you the range to expect. Every number in the worked examples and tables on this page was checked against an independent calculation in Python. If you spot a mistake, let us know and we'll correct it and note the change here.
References
- Karl Pearson, "On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen by random sampling", Philosophical Magazine, 5th series, 50, 1900, pages 157 to 175.
- Zacariah Labby, "Weldon's Dice, Automated", CHANCE, 22(4), 2009.
- William Feller, An Introduction to Probability Theory and Its Applications, Volume 1, 3rd edition, Wiley, 1968.
- Charles Grinstead and J. Laurie Snell, Introduction to Probability, 2nd edition, American Mathematical Society, 1997.
Related calculators
- Coin Flip Probability Calculator, for streaks, patterns and testing whether a coin is fair
- Card Probability Calculator, for draws where the deck changes after every card
- Binomial Probability Calculator, the general version of the repeated rolls maths
- Combinations and Permutations Calculator, which explains the C(n, k) term
- Probability Calculator, the full set of tools
Questions people ask
Why is 7 the most common total with two dice?
More combinations add up to 7 than to any other total: six of the 36 possible rolls. That gives 7 a 1 in 6 chance, against 1 in 36 for a 2 or a 12.
What are the odds of at least one 6 in several rolls?
Work out the chance of no 6 at all, which is (5/6) raised to the number of rolls, and subtract it from 1. For 4 rolls that gives 51.77%, not the 66.7% you get by adding 1/6 four times.
Are dice rolls independent?
Yes. Each roll of a fair die is unaffected by the ones before it. Expecting a number to be "due" after a long wait is called the gambler's fallacy.
How does advantage change my odds in D&D?
You roll two d20s and keep the higher, so you only fail if both miss. Needing 15 or higher goes from 30% to 51%, and a natural 20 goes from 5% to 9.75%.
How do I work out odds for dice with more than 6 sides?
The same methods work for any die. Enter the number of sides, or in Advanced mode type dice like 2d20 or 1d8+1d6. Two d20s have 400 equally likely combinations instead of 36.
What are the odds of rolling a Yahtzee?
Rolling five of a kind with five dice in a single roll is 6 in 7,776, which is 1 in 1,296 or about 0.0772%. In the game you get up to three rolls and can keep dice between them, which makes it far more likely.
Are real dice perfectly fair?
Not quite. In a famous 1894 experiment with 26,306 rolls of 12 dice, 5s and 6s came up slightly more often than they should, probably because scooped-out pips make those faces lighter. Casino dice fill in their pips to avoid this.