Dice Probability Explained — Bell Curves, Averages and Advantage
Two rolls can share the same range and behave completely differently. 1d12 and 2d6 both produce numbers up to 12. One of them is a coin flip dressed up as a die; the other is nearly predictable. Understanding the difference is the single most useful piece of maths in tabletop gaming, and it takes about ten minutes.
One die is flat
A single fair die is uniform: every face has exactly the same chance. On a d20 that is 5% per number, so a 3 is exactly as likely as a 17. On a d6 it is 16.7%. On a d100 it is 1%.
This is why a d20 system feels swingy. There is no gravity pulling results toward the middle. A character with a +9 bonus and a character with a +2 bonus roll the same shaped die, and the seven points between them get swamped by the nineteen points of spread the die itself contributes.
The average of a single die is (S + 1) ÷ 2. A d6 averages 3.5, a d8 averages 4.5, a d20 averages 10.5, a d100 averages 50.5. That formula is worth memorising because it makes every other calculation trivial.
Two dice make a curve
Roll two d6 and add them. There are 36 equally likely combinations, but only eleven possible totals, so the totals cannot be equally likely.
| Total | Ways to make it | 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% |
There is exactly one way to roll a 2 — both dice show 1 — and six ways to roll a 7. That makes 7 six times as likely as either extreme, and it is why craps is built around the number, why Catan puts its best payouts on 6 and 8, and why the middle of a Monopoly board is the busy part.
Add a third die and the pile gets steeper. On 3d6, results from 9 to 12 account for 48% of all rolls — nearly half of everything lands in four numbers out of sixteen. GURPS uses 3d6 precisely for that reason: it wants experts to be reliable and upsets to be rare.
The rule to take away: more dice means more predictable. Fewer, larger dice means more dramatic. Neither is better, and designers choose between them on purpose.
Adding dice averages
The average of any straight sum is just the sum of the averages.
3d6→ 3 × 3.5 = 10.58d6(Fireball) → 8 × 3.5 = 282d8+4→ 2 × 4.5 + 4 = 131d12→ 6.52d6→ 7
That last pair is the greataxe-versus-greatsword argument in fifth-edition DnD. The greatsword’s 2d6 averages half a point more than the greataxe’s 1d12, cannot roll below 2, and clusters near 7. The greataxe rolls a 12 one time in twelve rather than one time in thirty-six.
The Great Weapon Fighting style widens the gap further, and it is a nice illustration of why more dice is more of everything. The style lets you reroll 1s and 2s once. On a d6 that lifts the average from 3.5 to about 4.17; on a d12 it lifts 6.5 to about 7.33. So the greatsword gains twice — it has two dice to reroll — and ends at 8.33 against the greataxe’s 7.33. Half a point becomes a full point.
What advantage is actually worth
Advantage in DnD means rolling two d20s and keeping the higher. The average result rises from 10.5 to about 13.8, so people quote it as “worth about +3.3.”
That number is true and slightly misleading, because the benefit is nowhere near constant.
| You need | Normal | With advantage | Gain |
|---|---|---|---|
| 2+ | 95% | 99.75% | +4.75 |
| 6+ | 75% | 93.75% | +18.75 |
| 11+ | 50% | 75% | +25 |
| 16+ | 25% | 43.75% | +18.75 |
| 20 | 5% | 9.75% | +4.75 |
The maths behind it is simple: with advantage you fail only if both dice fail, so your failure chance gets squared. A 50% failure chance becomes 25%. A 5% failure chance becomes 0.25%.
The practical consequence is that advantage is worth most on the rolls that are genuine coin flips. Getting advantage on a check you were making 95% of the time is nearly wasted; getting it on the one that decides the fight is worth more than any bonus you could stack. Disadvantage is the mirror image — it takes 50% down to 25%.
Note also that advantage does not change the range at all. You can still roll a 1 with advantage; it just requires both dice to come up 1, which is a 1-in-400 event rather than 1-in-20.
Rerolls, drops and keeps
Drop lowest pushes the average up and shortens the tail. 4d6 drop lowest averages 12.24 rather than 4d6’s 14 or 3d6’s 10.5, and it makes an 18 possible but rare — about 1.6% per roll. Across six rolls you will see one about 9% of the time.
Rerolling low results raises the floor without touching the ceiling. A d6 that rerolls 1s once averages about 3.86; a d6 that rerolls 1s and 2s averages about 4.17. Neither can produce a 7.
Exploding dice do the opposite. If rolling maximum lets you roll again and add, a d6 averages 4.2 and has no ceiling at all — a very small chance of a very large number. Savage Worlds is built on this, and it gives even a d4 a long tail.
Three things people get wrong
“I’m due for a good roll.” Dice have no memory. After five 1s in a row, the chance of a 1 on the next roll is still exactly 1 in 20. Streaks are what randomness looks like — in 100 d20 rolls, seeing some number appear four times in a row is unremarkable.
“This die is cursed.” Possibly, but not on the evidence. To detect a genuinely biased die you need hundreds of rolls and a chi-squared test, not a bad evening. Cheap dice do have measurable imperfections, but they are far smaller than the run of bad luck you are attributing to them.
“Rolling 2d6 gives a number from 2 to 12, so each is about 9%.” This is the one that costs people games. The middle is many times more likely than the edges, and any decision made on the flat assumption is wrong.
Working it out at the table
Three formulas cover almost everything:
- Average of one die = (sides + 1) ÷ 2
- Average of a sum = add up the averages, then add the modifier
- Advantage = square your failure chance
For anything more complicated, roll it. The dice roller accepts full notation including keeps and drops, so you can try 4d6kh3 twenty times and watch the distribution appear rather than trusting an average you were told.