Dice Probability Calculator

What the dice probability calculator does

Rolling dice raises all sorts of probability questions, and this calculator handles the common ones. Choose the type of die and how many you are rolling, and it works out the chance of a particular sum, or of conditions like at least one die showing a certain value, or all of them clearing a threshold.

It covers dice from four to twenty sides and up to a hundred of them at once. Below is what it can answer and why dice sums behave the way they do.

How to use it

1. Choose the dice type, from 4-sided up to 20-sided, and the number of dice.
2. Pick the condition you want the probability of, then enter the value X it refers to.
3. Press Calculate for the probability and the odds, or Reset to clear it.

The questions it can answer

The conditions fall into two groups. Some are about the total across all the dice: the chance the sum equals a value, is at least a value, or is at most a value. These are the questions board games and tabletop games turn on.

The others are about the individual dice: the chance that at least one die shows a given value, that all of them do, or that they all land above or below some threshold. Between the two groups, the calculator covers most of the ways a roll of several dice can matter.

Why sums are not evenly likely

A single die is even-handed: on a six-sided die, every face from 1 to 6 is equally likely. But the moment you add dice together, that evenness disappears. With two dice, a sum of 7 is six times more likely than a sum of 2, because there are six different ways to make 7 and only one way to make 2.

The reason is that middle sums can be reached by many combinations, while the extremes can be reached by only one each. As you add more dice, this effect strengthens, and the distribution of the sum piles up in the middle into the familiar bell shape. So a question about a dice sum is really a question about how many combinations produce it, which is exactly what the calculator counts.

How the probabilities are worked out

For sums, the calculator counts the number of ways the dice can produce each total, building the count up one die at a time, and divides by the total number of possible rolls. That total is the number of sides raised to the power of the number of dice.

For conditions about individual dice, it often works through the opposite. To find the chance of at least one die showing a value, it finds the chance of none of them showing it and subtracts from certainty, which is far simpler than counting all the ways at least one could. Each condition rests on either counting combinations or turning the question around to its complement.

A worked example

Suppose you roll two six-sided dice and ask for the chance of a sum of 7. There are 36 equally likely rolls in total, six times six, and 7 can be made in six ways: 1 and 6, 2 and 5, 3 and 4, and those three reversed.

So the probability is 6 ÷ 36 = 16.67 percent, or 1 in 6. That makes 7 the most likely sum of two dice, which is why it is the number so many dice games are built around. A sum of 2 or 12, by contrast, has just one combination each, for a chance of only 1 in 36.

Entering your values

Choose the die type and the number of dice, select a condition, and enter the value X. For conditions about a single die's face, X should be a value the die can actually show. The calculator returns the probability as a percentage and as a "1 in" figure.

Questions people ask

What can this calculator work out?

The chance of a given sum across several dice, and conditions on individual dice, such as at least one showing a value or all of them clearing a threshold.

Why is 7 the most likely roll of two dice?

Because it can be made in six different ways, more than any other sum. The extremes, 2 and 12, have only one combination each.

Why do dice sums form a bell shape?

Middle sums can be reached by many combinations and the extremes by few. With more dice, this concentrates the sum in the middle into a bell-shaped distribution.

How is the chance of at least one die found?

By working out the chance that no die shows the value and subtracting from certainty. That is much simpler than counting every way at least one could show it.

References

A quick note on where the methods here come from. The probability rules for dice and sums are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering probability and combinations.

1. NIST/SEMATECH e-Handbook of Statistical Methods (probability). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (probability topics). https://openstax.org/details/books/introductory-statistics-2e