Start with one die
A standard die has six faces numbered 1 to 6. If the die is fair, each face is equally likely, so the chance of any one face is 1 in 6, about 16.7 percent. That is the whole foundation: list the possible outcomes, check that they are equally likely, and count how many of them you care about.
The basic formula is simple:
probability = number of favourable outcomes / number of possible outcomes
The chance of rolling an even number is 3 out of 6 (the faces 2, 4 and 6), which is 1/2. The chance of rolling a number greater than 4 is 2 out of 6 (5 and 6), which is 1/3. Probabilities always sit between 0 (impossible) and 1 (certain), and the probabilities of all the possible outcomes add up to 1.
Rolling a die more than once
Each roll of a fair die is independent. The die has no memory, so the result of the last roll does not change the next one. This is the reason the gambler’s fallacy is a fallacy: after five sixes in a row, the chance of a six on the next roll is still 1 in 6, assuming the die is fair.
Independence also gives the multiplication rule. To find the chance that two independent things both happen, multiply their probabilities. The chance of rolling a six twice in a row is 1/6 x 1/6 = 1/36, about 2.8 percent. Three sixes in a row is 1/216.
The same rule works for any specific sequence. The sequence 1, 2, 3 and the sequence 6, 6, 6 each have a chance of 1/216 with three dice. A sequence of sixes only feels special because we notice patterns. If you want the pattern-versus-chance question in more depth, the topic of random streaks is worth a read.
Two dice and the 36-outcome table
When you roll two dice, treat them as distinguishable, for instance one red and one blue. The first die has 6 results and the second has 6, so there are 6 x 6 = 36 equally likely outcomes. Writing them as a grid helps:
| Red \ Blue | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Each cell shows the sum, and each cell has a chance of 1/36. Counting how many cells show each total gives the distribution of sums:
| Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
The ways add up to 36, which is a useful check. A total of 7 can occur in 6 ways, so its chance is 6/36 = 1/6. A total of 2 or 12 can occur in only one way each, 1/36. The reason for the peak in the middle is covered in the article on why seven is the most common of the two dice totals.
A common mistake is to treat the sums themselves as equally likely, as if there were 11 outcomes and each had a 1 in 11 chance. They are not, because different sums can be made in different numbers of ways. Likewise, rolling a 1 and a 2 can happen two ways (1 then 2, or 2 then 1) while rolling a double 1 can happen only one way.
Using the table
Once you have the grid, many questions become counting exercises.
- Sum of at least 10: the cells 10, 11 and 12 have 3 + 2 + 1 = 6 ways, so 6/36 = 1/6.
- Doubles: the diagonal has 6 cells, so 6/36 = 1/6.
- A sum that is even: counting 1 + 3 + 5 + 5 + 3 + 1 = 18 ways, so 18/36 = 1/2.
- At least one die showing a six: 11 of the 36 cells, so 11/36, about 30.6 percent. This is easiest to find by the complement rule: 25 cells have no six, and 36 - 25 = 11.
That last example shows a shortcut that scales well. With many dice it is usually easier to find the chance that something does not happen and subtract from 1.
Three or more dice
With three dice there are 6 x 6 x 6 = 216 equally likely outcomes, and with four dice there are 1,296. A table stops being practical, but the same logic applies. For the sum of three dice, totals run from 3 to 18, and the middle totals of 10 and 11 are the most common, with 27 ways each out of 216 (12.5 percent each). The distribution becomes more bell-shaped as more dice are added, which is why many dice distributions in board games are designed the way they are.
For larger problems, tools such as combinations and permutations help with the counting, and the binomial distribution handles questions like “how many sixes in ten rolls”. When the counting gets heavy, a Monte Carlo simulation can estimate an answer by rolling virtual dice many times, though an estimate is only close to the exact value, not equal to it.
Fair dice and real dice
Everything above assumes a fair die, where every face is equally likely. Real dice are close to this, but small manufacturing imperfections can matter at a very fine level. If you suspect a die is biased, the approach is to count results over many rolls and compare them with what chance would produce.
A note on gambling
Dice games in casinos are built so the payouts are lower than the true odds suggest, which creates a house edge. Understanding the probabilities does not remove that edge, and no pattern of past rolls changes the chances of future ones. Gambling can be harmful; if it is causing you or someone you know difficulty, a local support service can help.