What is a dice probability calculator?
A dice probability calculator counts the ways a roll can make a total and divides by every equally likely outcome. With two six-sided dice, a total of 7 can be made 6 ways out of 36, so the chance is 16.7%, or 1 in 6. The calculator does the counting exactly for up to 12 dice with 2 to 100 sides, and shows the chance of rolling exactly the total, at least the total or at most the total.
Dice have no memory. Each roll is independent, so a total that has not come up for a while is no more likely next time. What the calculator can tell you is how likely a total is on one roll, and how likely it is to show up at least once in a number of rolls. That second figure is where intuition is often wrong.
How do you use the dice probability calculator?
- Enter the number of dice. The calculator accepts 1 to 12.
- Enter the sides per die. Six is standard. It accepts 2 to 100.
- Enter the total you want. This is the sum of all dice.
- Enter the number of rolls. It is used for the “at least once” line. It starts at 10.
- Read the large number. It is the chance of exactly that total on one roll. The panel gives the chance of at least or at most that total, the chance of seeing it in your number of rolls and a table of every total.
Example: 2 dice, 6 sides, total 7, 10 rolls. There are 6 ways to make 7 out of 36 outcomes. The chance on one roll is 16.7% (1 in 6). The chance of seeing a 7 at least once in 10 rolls is 83.8%.
How are dice probabilities calculated?
The number of outcomes is sides to the power of dice. The number of ways to make a total is found by building up the dice one at a time: for each die, every way to make the previous totals is added to the totals 1 to sides higher. This is a convolution, and the calculator does it with exact whole numbers. The chance of exactly the total is the ways divided by the outcomes. The chance of seeing it at least once in R rolls is 1 – (1 – p)^R, which treats each roll as independent.
| Step | Working | Result |
|---|---|---|
| All outcomes | 6 x 6 | 36 |
| Ways to make 7 | 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 | 6 |
| Chance of 7 | 6 / 36 | 16.7% |
| At least one 7 in 10 rolls | 1 – (30/36)^10 | 83.8% |
What does the full distribution of two dice look like?
The totals are not equally likely, because there are more ways to make the middle totals. Seven has the most ways (6) and 2 and 12 have the fewest (1). This shape is why games built on two dice, such as craps, treat 6, 7 and 8 so differently from 2 and 12.
| Total | Ways | Chance |
|---|---|---|
| 2 | 1 | 2.8% |
| 3 | 2 | 5.6% |
| 4 | 3 | 8.3% |
| 5 | 4 | 11.1% |
| 6 | 5 | 13.9% |
| 7 | 6 | 16.7% |
| 8 | 5 | 13.9% |
| 9 | 4 | 11.1% |
| 10 | 3 | 8.3% |
| 11 | 2 | 5.6% |
| 12 | 1 | 2.8% |
With three dice the pattern spreads out and the middle gets even heavier. The table shows all 16 totals out of 216 outcomes. Totals 10 and 11 can each be made 27 ways, a chance of 12.5% each, while 3 and 18 can be made only 1 way, a chance of 0.5%.
| Total | Ways | Chance |
|---|---|---|
| 3 | 1 | 0.5% |
| 4 | 3 | 1.4% |
| 5 | 6 | 2.8% |
| 6 | 10 | 4.6% |
| 7 | 15 | 6.9% |
| 8 | 21 | 9.7% |
| 9 | 25 | 11.6% |
| 10 | 27 | 12.5% |
| 11 | 27 | 12.5% |
| 12 | 25 | 11.6% |
| 13 | 21 | 9.7% |
| 14 | 15 | 6.9% |
| 15 | 10 | 4.6% |
| 16 | 6 | 2.8% |
| 17 | 3 | 1.4% |
| 18 | 1 | 0.5% |
How likely is a total to appear at least once?
A total with a 1 in 6 chance does not take six rolls to appear. The chance of at least one 7 grows with each roll but never reaches 100%.
| Rolls of two dice | Chance of at least one 7 |
|---|---|
| 1 | 16.7% |
| 3 | 42.1% |
| 6 | 66.5% |
| 10 | 83.8% |
| 20 | 97.4% |
| 36 | 99.9% |
This is the problem the gambler Antoine Gombaud, the Chevalier de Mere, put to Blaise Pascal in the 1650s. Betting on at least one six in 4 rolls of one die has a 51.77% chance, which is slightly better than even. Betting on at least one double six in 24 rolls of two dice sounds just as good, but its chance is 49.14%, slightly worse than even. It takes 25 rolls to reach 50.55%. The calculator lets you check this: enter 2 dice, 6 sides, total 12 and 24 rolls.
What does the calculator leave out?
- Biased dice. It assumes fair dice, where each face is equally likely.
- Game rules. It gives the chance of a total. It does not value a bet, re-rolls or the dice you keep.
- Order. It counts totals, not which die shows which number.
- Dependent rolls. Each roll is independent. Past rolls do not change the next one.
- Payouts. It does not tell you if a bet is fair. For that, compare the chance with the payout.
What mistakes do people make with dice odds?
- The gambler’s fallacy. A total that has not come up is not “due”.
- Treating all totals as equal. With two dice, 7 is six times more likely than 2.
- Adding chances. The chance of a 7 in two rolls is not 1/6 + 1/6. Use 1 – (5/6)^2, which is 30.6%.
- Forgetting the payout. A bet on a total pays less than the true odds in a casino, and that gap is the house edge.
- Mixing up “at least” and “exactly”. The panel shows each.
What should you try next?
Try 3 dice and a total of 10, then change to 2 dice and 12 with 24 and 25 rolls to see the Chevalier de Mere result. To see how a payout compares with the true odds, use the odds calculator and the expected loss calculator. For the casino dice game itself, see the craps odds calculator. If you want to set limits before you play, read the Safer play toolkit.
How to read the result
The large number is the chance of rolling exactly your total on one roll, with the ways and the "1 in" figure under it. The panel gives the chance of that total or more, that total or less, and the chance of seeing it at least once in your number of rolls. A table lists the ways and chance for each total near your target. The calculation assumes fair dice and independent rolls.
How it is calculated
Outcomes = sides^dice. Ways to make a total are found by adding one die at a time: for each die, every way to make a previous total is added to the totals 1 to sides higher (a convolution), using exact whole numbers. Chance of exactly the total = ways / outcomes. Chance of at least once in R rolls = 1 - (1 - p)^R.
Worked example
Worked example
2 dice, 6 sides, total 7, 10 rolls. Outcomes: 6^2 = 36. Ways to make 7: 6. Chance = 6 / 36 = 16.7% (1 in 6). At least one 7 in 10 rolls = 1 - (30/36)^10 = 83.8%. For 3 dice, a total of 10 has 27 ways out of 216, which is 12.5%.
Limits
The calculator assumes fair dice and independent rolls. It does not model loaded dice, re-rolls, kept dice, game rules or payouts, and it counts totals, not the order of dice. It works for 1 to 12 dice and 2 to 100 sides. It tells you how likely a total is, not whether a bet on it is good: for that, compare the chance with the payout. Results are exact for the inputs, and are not predictions of the next roll.