Dice Probability Calculator

Formula used

For one die, each face has probability 1 / s, where s is the number of sides. For multiple dice, the calculator uses dynamic programming to count how many outcome combinations produce each total.

The probability of a target condition is:

Probability = favorable outcomes / total outcomes

Total outcomes are:

Total outcomes = sⁿ

Here, n is the number of dice. A modifier shifts every total after the raw distribution is computed. The expected value is n × (s + 1) / 2 + modifier. The variance is n × (s² − 1) / 12.

How to use this calculator

Enter the number of dice you want to roll.

Set the number of sides on each die.

Add any modifier that changes the final sum.

Choose a mode such as exact, at least, at most, or between.

Enter the target sum or range values.

Press the calculate button to view the result above the form.

Review the chart, summary cards, and example data table.

Use the CSV or PDF buttons to export the current results.

Frequently asked questions

1. What does this calculator measure?

It measures the probability of reaching a chosen dice total or range. It also shows distribution details, expected value, variance, and cumulative probabilities.

2. Can I calculate probabilities for unusual dice?

Yes. You can enter any side count from 2 to 100. That includes common gaming dice like d4, d6, d8, d10, d12, d20, and many custom cases.

3. What does the modifier do?

The modifier shifts every final total after the dice are rolled. For example, a +2 modifier turns a raw total of 9 into a reported total of 11.

4. Why are some totals more likely than others?

Different totals can be formed by different numbers of combinations. Middle sums usually have more combinations, so they appear more often than extreme low or high sums.

5. What is cumulative probability?

Cumulative probability adds together probabilities up to a given total. It is useful when you want the chance of rolling at most a selected number.

6. Is the chart based on exact values?

Yes. The chart uses the full computed distribution from the calculator. Each plotted point represents the exact probability for one final sum.

7. When should I use exact versus between?

Use exact when only one total matters. Use between when a successful result can land anywhere inside a chosen lower and upper bound.

8. Does this tool work for board games and tabletop systems?

Yes. It works well for tabletop games, classroom probability exercises, simulations, and any scenario where you need precise total-sum odds.