Backgammon: The Chance of Rolling 2 Dice
This page gives a simple explanation of two-dice probabilities in backgammon. It also shows which opening rolls are most common, and which are hardest to roll.
Key point: a roll like 2 + 1 is not one of the hardest specific rolls. It is actually one of the most common specific combinations, because it can happen in two ways: 2 then 1 or 1 then 2.
1. How many possible rolls are there?
Each die has 6 faces. When you roll two dice, there are:
6 × 6 = 36 possible ordered outcomes
That means there are 36 equally likely results every time the dice are rolled.
Ordered outcomes means that 4 then 2 is different from 2 then 4.
2. Backgammon combinations
In backgammon, players normally treat 4 + 2 and 2 + 4 as the same roll value. So, instead of 36 ordered outcomes, we usually talk about 21 roll combinations:
- 15 non-doubles
- 6 doubles
3. Which specific rolls are most common?
Every specific non-double roll can happen in 2 ways.
For example, 4 + 2 can happen as:
- 4 then 2
- 2 then 4
So the chance of any specific non-double is:
2 out of 36 = 1 out of 18 = 5.56%
This means that all of these are equally common:
- 4 + 2
- 3 + 1
- 5 + 3
- 6 + 1
- 6 + 5
- 2 + 1
Any specific non-double
Example: 4 + 2, 3 + 1, 5 + 3, 6 + 1
5.56%
Any specific double
Example: 1 + 1, 4 + 4, 6 + 6
2.78%
4. Which specific rolls are hardest to get?
The hardest specific combinations to roll are the doubles, because each one can only happen in 1 way.
So the chance of any specific double is:
1 out of 36 = 2.78%
The least likely specific rolls are therefore:
- 1 + 1
- 2 + 2
- 3 + 3
- 4 + 4
- 5 + 5
- 6 + 6
So, 2 + 1 is not a worst-probability roll. It is one of the most common specific combinations at 5.56%. A double such as 4 + 4 is harder to roll at 2.78%.
5. What if you mean the total of the two dice?
Sometimes people mean the sum of the dice rather than the exact combination. This is different.
For totals, the rarest sums are:
- 2, which only happens as 1 + 1
- 12, which only happens as 6 + 6
Each has a probability of 1 out of 36 = 2.78%.
By contrast, the most common total is 7, which can be made in 6 ways.
6. Probability of each total
This chart shows how often each total appears when two dice are rolled.
| Total | Ways to roll it | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
7. The most common opening combinations in backgammon
The most common specific opening combinations are all the non-doubles. Each has the same chance of 5.56%.
Most common non-doubles
| Roll | Ways | Probability |
|---|---|---|
| 2 + 1 | 2 | 5.56% |
| 3 + 1 | 2 | 5.56% |
| 3 + 2 | 2 | 5.56% |
| 4 + 1 | 2 | 5.56% |
| 4 + 2 | 2 | 5.56% |
| 4 + 3 | 2 | 5.56% |
| 5 + 1 | 2 | 5.56% |
| 5 + 2 | 2 | 5.56% |
| 5 + 3 | 2 | 5.56% |
| 5 + 4 | 2 | 5.56% |
| 6 + 1 | 2 | 5.56% |
| 6 + 2 | 2 | 5.56% |
| 6 + 3 | 2 | 5.56% |
| 6 + 4 | 2 | 5.56% |
| 6 + 5 | 2 | 5.56% |
Least common specific rolls
| Roll | Ways | Probability |
|---|---|---|
| 1 + 1 | 1 | 2.78% |
| 2 + 2 | 1 | 2.78% |
| 3 + 3 | 1 | 2.78% |
| 4 + 4 | 1 | 2.78% |
| 5 + 5 | 1 | 2.78% |
| 6 + 6 | 1 | 2.78% |
8. Simple summary
- Two dice produce 36 ordered outcomes.
- In backgammon terms, these become 21 roll combinations.
- Any specific non-double such as 4 + 2 or 2 + 1 has a probability of 2/36 = 5.56%.
- Any specific double such as 3 + 3 has a probability of 1/36 = 2.78%.
- If you mean the total, then 2 and 12 are the rarest totals.
- The most common total is 7.
Prepared as a simple explanatory page for backgammon two-dice probabilities.