Dice look simple until you have to decide something with them: whether a board game rule is fair, how hard a target number should be, or whether rolling with advantage is worth giving up something else. The good news is that dice probability needs no statistics course. Every question below is answered the same way: count the equally likely outcomes, then count the ones you want. All the numbers in this article come from the exact counting in the dice and coin roller, which you can use to check any of them.
One die is flat, two dice are not
A fair six-sided die gives each face 1 time in 6. Nothing interesting happens until you add a second die. Two dice have 6 × 6 = 36 ordered outcomes (the first die and the second die are different objects, even if they look the same), and the totals are not equally common:
| Total | Ways | Chance |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
Seven peaks because it is the only total that every value of the first die can still reach: whatever the first die shows, exactly one face of the second die completes 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). For 12, only a 6 on the first die leaves any chance. That is the whole reason a board game with two dice treats 6 and 8 as good numbers and 2 and 12 as rare: 7 is six times as likely as 12.
Three dice make a bell
Add a third die and the triangle rounds into a bell. 3d6 has 6³ = 216 outcomes, and the counts run 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1 for the totals 3 to 18. Both 10 and 11 come up 27 times in 216, exactly 12.5% each, while 3 and 18 are 1 in 216, under half a percent.
This is why designers who want steady results use several small dice instead of one big one. A d20 and 3d6 both average 10.5, but a d20 gives 16 or more 25% of the time (5 faces out of 20) and 3d6 gives it only 4.63% (10 outcomes out of 216). A single die is swingy; a sum of several dice clusters around its middle.
Four dice, drop the lowest
Many role-playing games roll ability scores as 4d6 and keep the three highest (written 4d6kh3 or 4d6dl1). Now plain addition is not enough, because which dice count depends on all four. Counting still works: there are 6⁴ = 1296 outcomes, and for each one you drop the smallest die. The tool does this with a counting method instead of listing all 1296, and the result matches a brute-force enumeration:
- The average is exactly 15869/1296, about 12.24, against 10.5 for 3d6.
- The most likely score is 13 (172 outcomes, 13.27%).
- An 18 needs at least three sixes: four ways to place exactly three sixes, times five possible values for the other die, plus the single all-sixes roll, so 4 × 5 + 1 = 21 outcomes, 1.62%.
- A score of 15 or more comes up 23.15% of the time, against 9.26% for straight 3d6.
Dropping one die moves the whole curve up by almost two points, and it trims the bottom much more than it lifts the top: a score of 8 or less falls from 25.93% with 3d6 to 10.49%.
Advantage, worked exactly
With advantage you roll two d20 and keep the higher. The trick is to count failures instead of successes. To miss a target t, both dice must be below t, and each has a chance of (t − 1)/20 of being below it. So:
- chance with advantage = 1 − ((t − 1)/20)²
- chance with disadvantage = ((21 − t)/20)², because both dice must reach t
For a target of 11 that gives 1 − (10/20)² = 3/4 = 75% with advantage and (10/20)² = 25% with disadvantage, against 50% for one die. Here is the same calculation across the range:
| Need at least | One d20 | Advantage | Disadvantage |
|---|---|---|---|
| 5 | 80% | 96% | 64% |
| 8 | 65% | 87.75% | 42.25% |
| 11 | 50% | 75% | 25% |
| 15 | 30% | 51% | 9% |
| 18 | 15% | 27.75% | 2.25% |
| 20 | 5% | 9.75% | 0.25% |
The average with advantage is 13.825. The quickest way to see it: the expected value of a die is the sum, over every face k, of the chance that the result is at least k. For the higher of two d20 that is the sum of 1 − ((k − 1)/20)² for k = 1 to 20, which is 20 − 2470/400 = 13.825 (2470 is 0² + 1² + … + 19²). By symmetry, disadvantage averages 21 − 13.825 = 7.175.
Percentile dice
A d100 is usually two ten-sided dice: one marked 00, 10, 20 … 90 for the tens and one marked 0 to 9 for the units. You add them: 40 and 5 is 45, 00 and 7 is 7. The one rule people forget is that 00 and 0 together mean 100, not zero. With that rule every result from 1 to 100 has exactly one combination out of 100, so each is exactly 1%.
Common mistakes
- Treating 2d6 as a uniform 2 to 12. It is not: 7 is six times as likely as 2. A rule that triggers on "a roll of 2 or 12" happens 1 time in 18, not 2 times in 11.
- Counting 3+4 and 4+3 as one outcome. The dice are different objects. Counting unordered pairs gives 21 "outcomes" that are not equally likely, and every probability built on them is wrong.
- Calling advantage "+5". It is worth 25 percentage points only around a target of 10 or 11. At 20 it adds 4.75 points, less than +1 would. On average it is worth about 3.3.
- Assuming dice remember. After three 1s in a row the next roll is still 1 in 6 for each face. Streaks are expected in random sequences: in 10 coin flips a split of 7 and 3 or worse happens about a third of the time.
- Reading 00 + 0 as zero on percentile dice, which turns a 1 to 100 die into a 0 to 99 one and breaks every table built for d100.
- Adding averages of kept dice. The average of 4d6 keep 3 is not three quarters of 4d6's 14. It is 12.24, and only careful counting gives it.
Check it yourself
Type any expression into the dice and coin roller: 2d6, 3d6, 4d6kh3, d20 adv. The table under the result lists every total with its number of ways and its exact chance, and the target box gives "at least", "exactly" and "at most" for the number you need. Rolling uses the browser's cryptographic random source with rejection sampling, so the dice you roll there are as fair as the maths says they should be.