Backgammon odds: the chance to hit and to enter from the bar
Two dice have 36 possible rolls, all equally likely. 6-5 and 5-6 are two different rolls, but double six is one. Every chance in backgammon comes from that.
The chance to hit a lone checker
The distance is the number of points between your checker and the opponent’s, with no closed points in the way.
| Distance | Rolls that hit | Chance |
|---|---|---|
| 1 | 11 / 36 | 30.6% |
| 2 | 12 / 36 | 33.3% |
| 3 | 14 / 36 | 38.9% |
| 4 | 15 / 36 | 41.7% |
| 5 | 15 / 36 | 41.7% |
| 6 | 17 / 36 | 47.2% |
| 7 | 6 / 36 | 16.7% |
| 8 | 6 / 36 | 16.7% |
| 9 | 5 / 36 | 13.9% |
| 10 | 3 / 36 | 8.3% |
| 11 | 2 / 36 | 5.6% |
| 12 | 3 / 36 | 8.3% |
| 15 | 1 / 36 | 2.8% |
| 16 | 1 / 36 | 2.8% |
| 18 | 1 / 36 | 2.8% |
| 20 | 1 / 36 | 2.8% |
| 24 | 1 / 36 | 2.8% |
- The closer, the easier is not quite right: a checker 6 away is the easiest target (47%), because every roll with a 6 hits, and so do 5-1, 4-2, 3-3 and 2-2.
- Beyond 6 the chance drops sharply: from 7 away only 6 rolls hit (17%). That is why 7 or more counts as an indirect shot.
- Two blots at different distances add to the danger, but you can’t simply add the chances: one roll can hit either.
The chance to enter from the bar
| Closed points in the opponent’s home | You enter | You stay out |
|---|---|---|
| 0 | 100.0% | 0.0% |
| 1 | 97.2% | 2.8% |
| 2 | 88.9% | 11.1% |
| 3 | 75.0% | 25.0% |
| 4 | 55.6% | 44.4% |
| 5 | 30.6% | 69.4% |
| 6 | 0.0% | 100.0% |
The chance of not entering with one checker is the number of closed points times itself, out of 36. Against four closed points: 16 out of 36. With two checkers on the bar you need to bring both in, which is much harder.
More numbers worth remembering
| What | Chance |
|---|---|
| Any double | 6 / 36 = 16.7% |
| A given number on at least one die | 11 / 36 = 30.6% |
| A given non-double roll, such as 6-5 | 2 / 36 = 5.6% |
| A specific double, e.g. double sixes | 1 / 36 = 2.8% |
| Average pips per roll | 8.17 |
The dice here are completely random, and every roll can be checked at the end of the game: Locked dice.