Start by counting outcomes
Every dice question comes down to one fraction: the ways to get what you want, divided by all the ways the dice can land. A fair six-sided die has 6 equally likely faces, so the chance of any one face is 1 in 6, about 16.67%.
Add a second die and the outcomes multiply. Two dice give 6 x 6 = 36 combinations, and three give 6 x 6 x 6 = 216. Treat the dice as different colors when you count. A 1 on the red die and 2 on the blue is a different outcome from 2 on red and 1 on blue.
Formula notes
Probability = favorable outcomes / total outcomesTotal outcomes for N dice with S sides = S^N
Why 7 is the magic number with two dice
Only one combination makes 2 (1 and 1) and only one makes 12 (6 and 6). Seven can be made six ways: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. So 7 shows up 6 times in 36, while 2 and 12 each show up once.
The odds climb by one combination per step from 2 up to 7, then fall back the same way to 12. That's the triangle shape in the 2d6 table on our dice roller page. Six and 8 come close to 7 at 5 ways each, while the ends of the range are rare.
- 2 or 12: 1 way each, 2.78%
- 3 or 11: 2 ways each, 5.56%
- 4 or 10: 3 ways each, 8.33%
- 5 or 9: 4 ways each, 11.11%
- 6 or 8: 5 ways each, 13.89%
- 7: 6 ways, 16.67%
"At least one" questions
People often guess that two dice give a 2 in 6 chance of at least one 6. It's lower, because the case where both dice show 6 gets counted twice. The clean way is to work out the chance of missing every time and subtract it from 1.
With two dice, each one misses a 6 with probability 5/6. Both miss with probability 25/36, so at least one 6 shows up 11 times in 36, about 30.56%. With four dice it's 1 - (5/6)^4, about 51.77%, just better than a coin flip.
Formula notes
Chance of at least one = 1 - (chance of missing on one die)^N
Three dice and the bell curve
Roll 3d6 and the totals run from 3 to 18 across 216 outcomes. The middle gets crowded fast. Ten and 11 can each be made 27 ways, a 12.5% chance, while 3 and 18 can each be made only one way, about 0.46%.
The more dice you add, the more the totals bunch around the average and the rarer the extremes become. That's why a game that rolls 3d6 for a score produces far more 10s and 11s than 3s or 18s, while a single d20 makes every number from 1 to 20 equally likely at 5%.
Rolling twice and keeping one
Some games let you roll a d20 twice and keep the higher result. To succeed you only need one of the two rolls to hit, so the chance is 1 minus the chance both miss. If you need 11 or higher, each roll misses half the time, both miss a quarter of the time, and you succeed 75% of the time instead of 50%.
Keeping the lower roll flips it. Both rolls must succeed, so the chance is squared: 0.5 x 0.5 = 25%. The boost is biggest for middle targets. At a target of 20 it lifts the chance only from 5% to 9.75%.
The average roll and the spread
The average of one fair die is (sides + 1) / 2. A d6 averages 3.5 and a d20 averages 10.5. For several dice, multiply by the number of dice: 2d6 averages 7 and 3d6 averages 10.5, the same as a single d20.
Those two averages hide very different spreads. A d20 can land anywhere from 1 to 20 with equal odds, while 3d6 lands on 9, 10, 11 or 12 about 48% of the time (104 of 216 outcomes). The standard deviation puts a number on it: about 2.42 for 2d6, 2.96 for 3d6 and 5.77 for 1d20. Pick 3d6 when you want steady results and a d20 when you want swingy ones.
Mistakes that trip people up
The gambler's fallacy is the big one. Dice have no memory, so a 7 is just as likely after five 7s in a row as after none. Another is reading a small sample as proof of bias. Twenty rolls can easily look lopsided on a perfectly fair die. Finally, averages describe the long run. A 2d6 roll averages 7, but any single roll can land anywhere from 2 to 12.
Frequently asked questions
How many combinations are there with 3 dice?
216. Each die has 6 faces, so three dice give 6 x 6 x 6 = 216 equally likely ordered outcomes.
What is the probability of rolling a 7 with two dice?
6 in 36, or 1 in 6, which is about 16.67%. No other total is as likely.
What is the most common total with 3d6?
10 and 11 tie as the most common totals. Each can be made 27 ways out of 216, a 12.5% chance.
Does rolling twice and keeping the higher roll double your chances?
No. It raises a 50% chance to 75%, and a 5% chance only to 9.75%. The gain is largest for middle targets.
Can a die be 'due' for a number?
No. Each roll of a fair die is independent, so past results don't change the next one.
References
These sources support the method or guidance used for Dice Roller. Verify time-sensitive rules at the source.
Try the calculator
Open Dice Roller, enter your scenario, and compare its supporting rows with this guide's method and checks.