Why seven turns up so often
Calculates the chance of rolling a given sum on any number of dice and shows the full distribution of totals.
One die is flat: every face is equally likely. Two dice are not, because there are six ways to make seven and only one to make twelve, and that asymmetry is what every dice game is built on.
Add more dice and the distribution narrows into a bell. Three six-sided dice almost never total four or seventeen, and that is what makes 3d6 a usable stand-in for an average.
What this calculator shows
- The probability of the sum you ask for, exactly, at least or at most
- The average roll and the commonest total
- The full distribution of every possible sum
What to keep in mind
- The dice are fair and independent, and the sum is the plain total with no rerolls or modifiers.
- The distribution is computed exactly rather than simulated, so the percentages are the true values.
FAQs
What is the chance of rolling 7 on two dice?
One in six, or 16.67 percent. It is the commonest total because six different pairs of faces make it.
What is the average roll?
Half the number of faces plus a half, times the number of dice. For 2d6 that is 7; for 3d6 it is 10.5.
Why do more dice narrow the spread?
Because extreme totals need every die to agree, and that gets rapidly less likely. It is the central limit theorem showing up on a table.
Does 2d6 differ from 1d12?
Completely. A d12 gives every number from 1 to 12 equal weight; 2d6 makes 7 six times as likely as 12 and cannot produce a 1 at all.
Worked example
Seven on two dice
Input: 2 dice, 6 faces, exactly 7
Output: 16.67 percent
Note: Six of the 36 outcomes make seven, against one for twelve. That six-to-one gap is why craps pays what it pays.