Two dice are not one bigger die

Rolling 2d6 gives totals from 2 to 12, and rolling a single eleven-sided die would give 1 to 11 — the same span, and a completely different game. The two dice produce a 7 on 16.67 per cent of rolls against 9.09 per cent for any single face of the flat die, because there are six ways to make 7 from two dice and only one way to make 2.

That hump in the middle is why board games use multiple dice. A flat distribution treats every outcome as equally surprising; a curve makes the middle ordinary and the extremes memorable.

What does the 2d6 curve look like?

Thirty-six equally likely combinations, distributed by how many ways each total can be made.

Seven is six times more likely than twelve. Anything from 6 to 8 covers 44.4 per cent of rolls, and the two extremes together account for 5.6 per cent.

What happens with three dice?

The curve gets steeper. With 3d6 there are 216 combinations, the mean is 10.5, and the two central totals of 10 and 11 alone take 25 per cent of all rolls — against 12.5 per cent for the equivalent pair on a flat sixteen-sided die.

The tails get thin quickly. Rolling 15 or more on 3d6 happens 9.26 per cent of the time, and rolling an 18 happens once in 216. Adding dice pulls results toward the middle, which is the central limit theorem visible on a table.

Why does 4d6 drop lowest exist?

Because it shifts the average upward without changing the range. Rolling four six-sided dice and discarding the lowest gives a mean of 12.24 against 10.5 for straight 3d6, and it raises the chance of a 15 or better from 9.26 per cent to 23.15 per cent.

The range is identical — 3 to 18 either way — so the change is entirely in the shape. It is the standard method for generating characters in several role-playing systems for exactly that reason: same possible outcomes, better expected ones, and the extremes still reachable.

The effect is general. Any sum of independent dice tends toward a bell shape as you add more of them, and it does so quickly — three dice already look distinctly humped, and five are close to a smooth curve. The width grows with the square root of the number of dice while the range grows linearly, so more dice means a proportionally tighter result.

What does the notation actually mean?

A count, the letter d, and the number of faces, with optional modifiers and groups.

• 2d6 — two six-sided dice, totalled.
• 3d8+2 — three eight-sided dice, plus two.
• d20 — one twenty-sided die; the leading 1 is optional.
• 4d6-1+d4 — groups combine, and each die is rolled independently.
• d100 — percentile, either as a hundred-faced die or two ten-sided ones.

The modifier is added once to the total, not to each die. That distinction matters: 3d6+3 has a range of 6 to 21 with the same curve shifted up, while 3d(6+1) would be a different die entirely.

Groups combine before the modifier, which is what makes the notation compact enough to write in a rulebook. A roll written as 2d6+1d4+3 draws three separate dice and adds a constant, and the resulting curve is the sum of two different humps offset by three — not something anyone would want to describe in words.

Which shape do you actually want?

It depends on whether you want surprise or texture. A flat die makes every outcome equally likely and every result equally plausible, which suits a lookup table where each entry should come up as often as any other. Multiple dice make the middle common and the edges rare, which suits anything modelling a skill or an average.

The practical test: if a critical failure and an average result should be equally frequent, use one die. If a critical failure should be a story, use several.

Questions people ask

Are the dice independent? Yes. Each die is drawn separately, so a 6 on the first tells you nothing about the second. The curve comes from counting combinations, not from any link between the dice.

Can I roll dice that do not exist? In notation, yes — d7, d13, d1000 are all well defined even where no physical die has that shape. Digital rolling has no manufacturing constraint.

Why does the roller highlight the maximum? Because a natural maximum is the result people are looking for, and finding it in a list of eight numbers is slower than it should be.

Is a physical die fair? Close, and not exactly. Pipped dice have slightly unequal face weights because the pips remove material, which is why casino dice have flush, filled pips.

Why does d100 sometimes use two dice? Because a hundred-faced die is impractical, so percentile is rolled as two ten-sided dice, one read as tens and one as units. The distribution is flat either way — it is two digits of one number rather than a sum of two rolls.

Pick the number of dice for the shape you want, not the range. The dice roller accepts full notation and keeps a history, the random number generator gives the flat distribution instead, and the coin flip is the two-sided edge case of the same idea.