Dice Roller

Dice Roller: How It Works

One die gives every face an equal chance. Two dice do something quite different — the sum has a triangular distribution where 7 is six times more likely than 2. That difference is the basis of most dice-based game design.

Two six-sided dice

Sum	Ways	Probability
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 are 36 equally likely outcomes and six of them sum to 7, against one that sums to 12. This is why board games place their most valuable resources on 6, 7 and 8, and why the odds in dice games are structured as they are.

Dice notation

Tabletop games use a compact notation: NdS+M, where N is the number of dice, S the number of sides and M a modifier.

Notation	Means	Range	Average
1d6	One six-sided die	1–6	3.5
2d6	Two six-sided dice	2–12	7
3d6+2	Three dice plus 2	5–20	12.5
1d20	One twenty-sided die	1–20	10.5
4d6 drop lowest	Roll four, keep best three	3–18	≈ 12.24

The average of a single die is (sides + 1) ÷ 2, and averages add — so 3d6 averages 10.5.

One die or several

This is the key design choice. 1d20 gives a flat distribution: every outcome equally likely, so results swing wildly. 3d6 covers a similar range but clusters heavily around the middle, so extreme results are rare.

A game using 1d20 has high variance — a novice can beat an expert on a lucky roll. A game using 3d6 rewards the modifier more, because the roll itself is predictable. Neither is better; they produce different feels.

Advantage and disadvantage

Rolling two dice and taking the higher shifts the average of a d20 from 10.5 to about 13.8, and taking the lower shifts it to about 7.2. It is a large effect achieved without changing any number on the character sheet, which is why the mechanic is popular — it is roughly equivalent to a +5 or −5 modifier but has different effects at the extremes.

Are physical dice fair?

Standard moulded dice have hollowed pips, making each face slightly different in weight, and inexpensive dice are frequently measurably biased. Casino dice are machined to tight tolerances with flush inlaid pips precisely to avoid this. A digital roll using a cryptographic random source is fairer than almost any physical die.

The classic fairness test is the salt-water float: a fair die suspended in dense salt water tumbles randomly, while a biased one repeatedly settles with the same face up.

Frequently Asked Questions

Why is 7 the most common roll with two dice?

Because six of the 36 possible combinations sum to 7, more than any other total. Only one combination gives 2 and one gives 12, making 7 six times as likely as either.

What does 3d6+2 mean?

Roll three six-sided dice, add the results, then add 2. The range is 5 to 20 with an average of 12.5. This NdS+M notation is standard across tabletop games.

What is the difference between rolling 1d20 and 3d6?

1d20 is flat — every result equally likely, producing wild swings. 3d6 clusters around the middle, making extremes rare and modifiers more decisive. They cover similar ranges and feel completely different.

How much does advantage change the odds?

Rolling two d20 and taking the higher raises the average from 10.5 to about 13.8, roughly equivalent to a +5 modifier — though it behaves differently at the extremes, helping most when you need a middling result.

Are physical dice actually fair?

Often not exactly. Hollowed pips make faces differ slightly in weight, and cheap dice are frequently measurably biased. Casino dice use flush inlaid pips and tight tolerances specifically to avoid this.

Is a digital roll truly random?

It uses your browser's cryptographic random source, which is fairer than almost any physical die and cannot be predicted. Each roll is independent of every previous one.