Calculating dice odds
Dice games and tabletop role-playing games are full of probability questions: what is the chance of rolling at least one six, of getting a double, or of every die showing the same face? This calculator handles any number of dice from four-sided to twenty-sided, and gives the probability that at least k dice, exactly k dice or no die shows a chosen face, plus the average and spread of the total.
The formulas
With n dice of s sides, the chance that one die shows a given face is 1/s. The number of dice showing it follows a binomial distribution:
- No die shows it: ((s − 1) ÷ s)^n
- At least one: 1 − ((s − 1) ÷ s)^n
- Exactly k: C(n, k) × (1/s)^k × ((s − 1)/s)^(n − k)
- All dice the same face: s^(1 − n)
- Average total: n × (s + 1) ÷ 2; standard deviation: √(n × (s² − 1) ÷ 12)
The same formulas apply to rolling one die n times.
Worked example: the Chevalier de Méré’s bet
In 17th century Paris, the gambler Antoine Gombaud, known as the Chevalier de Méré, made money betting that he would roll at least one six in four rolls of a die.
- Chance of no six in four rolls: (5/6)⁴ = 48.2%.
- Chance of at least one six: 1 − 0.482 = 51.8%, a small but steady edge.
- He then bet on at least one double six in 24 rolls of two dice, reasoning by proportion. The real chance is 1 − (35/36)²⁴ = 49.1%, and he started losing.
- His question to Blaise Pascal led to Pascal’s letters with Pierre de Fermat in 1654, which are often seen as the birth of probability theory.
At least one six by number of dice
| Dice or rolls | At least one six | No six |
|---|---|---|
| 1 | 16.7% | 83.3% |
| 2 | 30.6% | 69.4% |
| 3 | 42.1% | 57.9% |
| 4 | 51.8% | 48.2% |
| 6 | 66.5% | 33.5% |
| 10 | 83.8% | 16.2% |
| 24 | 98.7% | 1.3% |
Useful benchmarks
- Two dice: a total of 7 is the most likely, 6 chances in 36 (16.7%). A double is also 1 in 6.
- Five of a kind with five dice in a single throw: 6 ÷ 6⁵ = 1 in 1,296.
- A natural 20 on a d20: 5%; with advantage (roll two, keep the higher), 9.75%.
- Four dice average 14, with a standard deviation of 3.4, so most totals land between about 10 and 18.
To work with sums of dice in detail, remember that the total of several dice is not uniform: middle values are far more likely than extremes, and with many dice the distribution approaches a bell curve.