Backgammon Probability
Dice decide every turn, but they are not a mystery. Strong players do not get luckier rolls; they arrange their checkers so that more of the possible rolls work for them. All of that rests on one simple count.
Why 36 rolls
Two dice give 6 × 6 = 36 combinations, and each is equally likely. The key detail is that the two dice are different objects. A roll of 5-3 can come up as 5 on the first die and 3 on the second, or the other way around, so it accounts for 2 of the 36 combinations. A double such as 4-4 can happen only one way.
- Any particular non-double (for example 6-1): 2/36, about 5.6%.
- Any particular double (for example 6-6): 1/36, about 2.8%.
- Any double at all: 6/36, about 16.7%.
- At least one specific number on either die: 11/36, about 30.6%.
The last line surprises beginners. It is not 12/36, because the roll where both dice show that number would be counted twice.
Hitting a blot
To work out the chance of hitting, count every roll that lets a checker travel the needed distance. A shot of six points or less is a direct shot: it can be hit with a single die, with a combination of both dice, or with a double. A shot of seven or more is indirect and needs both dice. The table assumes no blocked points in between; every enemy point in the way removes some of these rolls.
| 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% |
Two lessons follow. First, a blot seven or more points away is far safer than one within six. Second, beyond twelve points only doubles can hit: 15, 16, 18, 20 and 24 points can each be reached by exactly one double, so each is a 1/36 shot, and other distances above 12 cannot be hit at all in one roll.
Entering from the bar
When you are hit, your checker needs an open point in your opponent's home board. Each closed point removes options, and the odds fall quickly:
| Points closed | Rolls that enter | Chance |
|---|---|---|
| 1 | 35/36 | 97.2% |
| 2 | 32/36 | 88.9% |
| 3 | 27/36 | 75.0% |
| 4 | 20/36 | 55.6% |
| 5 | 11/36 | 30.6% |
With all six points closed, no roll can enter and the player on the bar simply waits. This is why a closed board is so powerful and why making home board points matters so much in basic strategy.
Pips and the race
Each player starts with 167 pips to travel. Counting doubles as four moves, the average roll is worth 8.17 pips (49/6). Comparing pip counts tells you who leads the race and by how many rolls, which feeds directly into decisions about the doubling cube.
You do not need to memorize every number. Learn the direct shot figures and the entering table first; most practical decisions grow out of those two lists. Our backgammon tips show how to put them to work over the board.