Backgammon odds: the numbers worth knowing
Two dice make 36 equally likely rolls. Count how many of them do something and you have its odds. Two tables cover most of what you need over the board: how often a blot gets hit, and how often a checker on the bar comes in.
Counting rolls
- 5-2 and 2-5 are different rolls, so there are 36, not 21. Each non-double is 2 rolls in 36; each double is 1.
- Any double: 6 in 36, 1 in 6.
- A given number on at least one die (say a 6): 11 in 36, about 31%.
How often a blot gets hit
A blot is a single checker. Its risk depends on how far behind it your opponent's checkers are. The table is for one opposing checker with nothing in the way: points your side holds in between take some rolls away, and each extra checker in range adds some.
| Pips away | Rolls of 36 | Chance |
|---|---|---|
| 1 | 11 | 31% |
| 2 | 12 | 33% |
| 3 | 14 | 39% |
| 4 | 15 | 42% |
| 5 | 15 | 42% |
| 6 | 17 | 47% |
| 7 | 6 | 17% |
| 8 | 6 | 17% |
| 9 | 5 | 14% |
| 10 | 3 | 8% |
| 11 | 2 | 6% |
| 12 | 3 | 8% |
Past 12 only doubles reach: 15, 16, 18, 20 and 24 pips away are 1 roll in 36 each, and 13, 14, 17, 19, 21, 22 and 23 cannot be hit in one roll.
What to take from it: within 6 pips (a direct shot) a blot is hit 11 to 17 times in 36, about 31% to 47%. From 7 to 12 pips (an indirect shot, reached only by combining both dice or playing a double three or four times) it is 6 or fewer. So when you have to leave a blot, leave it 7 or more pips from your opponent's checkers. The worst distance is exactly 6.
Two blots are not twice the risk: some rolls hit either one. Count the rolls that hit at least one, and count each roll once.
Coming in from the bar
A checker on the bar enters on a die that matches an open point in your opponent's home board. It stays out only when both dice show closed points.
| Closed points | One checker enters | Both of two enter |
|---|---|---|
| 0 | 36 of 36 (100%) | 36 of 36 (100%) |
| 1 | 35 of 36 (97%) | 25 of 36 (69%) |
| 2 | 32 of 36 (89%) | 16 of 36 (44%) |
| 3 | 27 of 36 (75%) | 9 of 36 (25%) |
| 4 | 20 of 36 (56%) | 4 of 36 (11%) |
| 5 | 11 of 36 (31%) | 1 of 36 (3%) |
| 6 | 0 of 36 (0%) | 0 of 36 (0%) |
The drop is steep: against a 4-point board one checker still comes in more than half the time, against a 5-point board less than a third. This is why closing points in your home board before you hit pays off. Against all six you cannot enter at all until a point opens.
Odds and Tavla
The dice are the same in Tavla, so are the numbers. Tavla has no doubling cube; a mars counts 2, like a gammon. See Tavla rules.
Questions people ask
What are the odds of rolling a double in backgammon?
1 in 6: six of the 36 rolls are doubles.
How likely is a blot 6 pips away to be hit?
17 rolls in 36, about 47%, on an open board. It is the most likely distance.
How often do I come in against a 5-point board?
11 rolls in 36, about 31%: only rolls that show the one open number.