# Free online dice roller and probability calculator

> A dice roller that simulates any combination of polyhedral dice and shows the statistical range of possible results. Enter a target number to see the probability of rolling that.

Roll any combination of dice (d4, d6, d8, d10, d12, d20, d100) with a modifier, and calculate the probability of rolling at or above a target number.

**Interactive version:** https://onlinecalculator.me/everyday/dice/
**Category:** Everyday

## How to use

1. Choose the number of dice (1–20) and the die type (d4 through d100).
2. Enter a modifier to add or subtract from the total.
3. Optionally enter a target number to see the probability of rolling that or higher.
4. Click **Roll dice** to see your result and statistics.

## Formulas

```
min  = numDice × 1 + modifier
max  = numDice × sides + modifier
mean = numDice × (sides + 1) / 2 + modifier

variance per die = (sides² − 1) / 12
total variance   = numDice × variance per die
stdDev           = √(total variance)
```

Single-die probability (exact):
```
P(roll ≥ target) = (sides − (target − modifier) + 1) / sides
```

Multi-die probability (normal approximation with continuity correction):
```
P(sum ≥ target) ≈ P(Z ≥ (target − 0.5 − mean) / stdDev)
```

## Worked example

Rolling **2d6+3** with a target of 10:

```
min  = 2×1+3  = 5
max  = 2×6+3  = 15
mean = 2×3.5+3 = 10
variance = 2×(36−1)/12 = 5.833
stdDev = √5.833 ≈ 2.415

P(roll ≥ 10) ≈ P(Z ≥ (10 − 0.5 − 10) / 2.415) = P(Z ≥ −0.207) ≈ 58%
```

## Notes

- The normal approximation is most accurate for 3 or more dice. For a single die, exact probabilities are always used.
- The d100 (percentile die) is equivalent to rolling 1d100 and produces results from 1 to 100.
- Negative modifiers can push the minimum result below zero.

## Frequently asked questions

### How is the mean of a dice roll calculated?

The mean of rolling n dice with s sides is n × (s + 1) / 2. For 2d6 that is 2 × 7 / 2 = 7. Adding a modifier shifts the mean by the same amount.

### How is the probability of reaching a target calculated?

For a single die, the probability of rolling ≥ target is (sides − target + 1) / sides — an exact calculation. For multiple dice, the calculator uses a normal approximation with continuity correction, since summing dice distributions approaches a normal curve.

### What is a modifier?

A modifier is a flat number added to or subtracted from the total. In tabletop RPGs like D&D, a +3 modifier means every roll is 3 higher than the dice alone. Modifiers shift the min, max, and mean by the same fixed amount.

### What does standard deviation mean for dice?

Standard deviation measures how spread out results are. A larger standard deviation means rolls vary more from the mean. For n dice with s sides, the variance per die is (s² − 1) / 12, and the total variance is n times that.

### How do I share my dice configuration?

Click Share to copy the page link, or Share with my numbers to include your dice settings in the URL.

## Sources

- [Dice](https://en.wikipedia.org/wiki/Dice) — Wikipedia
- [Dice (probability of sums)](https://mathworld.wolfram.com/Dice.html) — Wolfram MathWorld

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