Three Dice Cluster, and One Die Does Not

Why the two systems cross

Three dice have many ways to make a middling total and very few to make an extreme one — 216 combinations in all, and only one of them is three sixes. A single die has exactly one way to make each of its faces, so nothing is rarer than anything else. The consequence for play is that a bell curve rewards competence and punishes ambition: a modest bonus moves a 3d6 roll a long way when the target is near the middle and almost nowhere when it is out at the edge, while the same bonus on a d20 is worth a flat five percentage points per point wherever the target sits.

Exact beats simulated, and it is not close

A simulation answers by rolling many times and counting, so its answer wobbles: ten thousand rolls of 3d6 gave 9.190 % for at least 15 against a true 9.259 %, and a second run would land somewhere else. Convolution answers by combining the distributions directly — every outcome of the first die against every outcome of the second, then that against the third — and gives the exact fraction every time. For dice the exact route is also faster, which is why there is no reason to simulate: three six-sided dice have 216 outcomes, and enumerating 216 things is instant.

Frequently asked questions

Which system should a game use?

It depends on how often you want the unlikely to happen. A flat die makes upsets common — a novice beats a master one time in twenty at the same target — which suits a game that wants swings. A bell curve makes them rare and makes bonuses matter, which suits a game where a trained character should reliably do trained things. Neither is better; they are different promises to the table, and it is worth making the promise on purpose.

Does the calculator handle dice other than d6?

Yes — any number of dice with any number of faces, plus a flat modifier, and the same convolution applies. The counts grow quickly (five d10 have a hundred thousand outcomes) but the method does not care, because it combines distributions rather than enumerating rolls. Keep the roll notation the same as your rules text and the answer will match the table.

What about advantage — rolling twice and taking the best?

That is a different calculation and it is worth doing separately, because its benefit is not constant. Taking the best of two d20 helps most in the middle of the range and least at the extremes: against a target you were going to hit anyway, or one you were never going to hit, the second roll changes little. The peak is around a fifty-fifty target, where it adds roughly twenty-five points.