Backgammon Probability Basics
Every backgammon decision is secretly a probability question: how likely am I to be hit, to enter from the bar, to win this race? The good news is that the entire mathematical foundation of the game fits on one page — this one.
The 36 Combinations
Two dice produce 36 equally likely combinations (6 × 6). Because 6-2 and 2-6 play identically, players talk about 21 distinct rolls — 15 mixed rolls and 6 doubles — but every probability in the game comes from counting out of 36:
- Each mixed roll (like 6-5) occurs 2 ways out of 36 — about 5.6%.
- Each double occurs exactly 1 way out of 36 — about 2.8%.
- Any double at all: 6/36 = 1 in 6 per roll.
- At least one specific number (say, at least one 6): 11/36 — the 6-x rolls both ways (10) plus 6-6 (1).
That last number is the workhorse: any single number you need — to hit, to enter, to escape — appears on 11 of 36 rolls, just under one third of the time. Want to see the grid in action? Our fair dice roller throws with cryptographic randomness and tallies how often doubles actually arrive.
Hitting Odds at Every Distance
How likely is your blot to be hit? Count the pips between it and each enemy checker and read the table. These are the classic unblocked numbers — made points in between can only lower them, because they kill the combination rolls.
| Distance (pips) | Rolls that hit | Chance |
|---|---|---|
| 1 | 11/36 | 31% |
| 2 | 12/36 | 33% |
| 3 | 14/36 | 39% |
| 4 | 15/36 | 42% |
| 5 | 15/36 | 42% |
| 6 | 17/36 | 47% |
| 7 | 6/36 | 17% |
| 8 | 6/36 | 17% |
| 9 | 5/36 | 14% |
| 10 | 3/36 | 8% |
| 11 | 2/36 | 6% |
| 12 | 3/36 | 8% |
| 15 / 16 / 18 / 20 / 24 | 1/36 each | 3% |
Two lessons jump out. First, direct shots (distance 6 or less) are all dangerous — a third to nearly half — and 6 is the single most dangerous distance at 17/36, because the 11 rolls containing a 6 are joined by the sums: 5-1 and 4-2 (both ways each), 3-3, and even 2-2 (three deuces). Second, the cliff between 6 and 7 pips is enormous: 47% falls to 17%. When you must leave a blot, leaving it 7 or more pips away instead of 6 or fewer is often the whole difference between a safe game and a lost one.
Entering From the Bar
The same 11/36 logic drives re-entry. Each open point in your opponent's home board gives you entering numbers; with n points open your chance of entering with at least one checker is:
- 5 points open: 35/36 (97%)
- 4 points open: 32/36 (89%)
- 3 points open: 27/36 (75%)
- 2 points open: 20/36 (56%)
- 1 point open: 11/36 (31%)
This is why an early hit is cheap (the opening board has five open points) and a late hit against a strong board is catastrophic — and why building home-board points before you attack is not pedantry but arithmetic. The mechanics of the bar are in the rules reference.
The Pip Count: Measuring the Race
The pip count is the total number of pips a player still needs to bring all fifteen checkers home and bear them off. Multiply each checker's point number by the checkers on it and add everything up. From the starting position both players count 167: 2×24 + 5×13 + 3×8 + 5×6.
The count turns "who's winning?" into a number:
- Clearly ahead (say 10%+ of your count): break contact and race — every exchange of hits can only help the trailer.
- Clearly behind: don't race. Keep an anchor, keep contact, play for the hit.
- Close race: position decides — structure, blots and home boards matter more than a few pips.
You don't need a full recount every turn. Count once, then update incrementally: your roll subtracts its pips from your count, a hit adds the full return distance to the victim's. Getting hit on your opponent's 5-point costs 20 pips; getting hit next to home costs a fortune — probability and the race are the same subject. To practice, set up any position in our pip count calculator, guess both counts, then check yourself.
Doubles Change the Math
Doubles play each number four times, so 6-6 moves 24 pips — the biggest single throw in the game — while the average roll moves about 8.17 pips. One roll in six being a double is also why races are never quite over: a trailer within a couple of big doubles always has real chances, which feeds directly into cube decisions and the value of gammons.
A Worked Example
Suppose you must leave one blot, and the choice is a checker 5 pips in front of an enemy point or one 8 pips in front of it. The table says 15/36 against 6/36 — the "safer-looking" close blot is two and a half times more likely to be hit. Now add context: if your opponent's home board has four points made, being hit likely costs the game, so the 8-pip play is nearly mandatory; if their board is wide open, being hit costs a re-entry and a few pips, and a bolder play that builds your position can be right. That two-step habit — read the raw odds, then weigh the price of the bad outcome — is the entire method, and it applies to hits, splits, slots and cube decisions alike.
Using the Numbers at the Table
Nobody computes at the board; strong players just recognize the handful of thresholds: any number is 11/36, direct shots are dangerous and 6 is the worst, 7+ pips is the safety cliff, doubles are 1 in 6, the race starts at 167. Combine those with the beginner habits and the rollout-approved openings, and your "luck" will improve remarkably. Then test it where the dice are provably honest — play free in your browser with certified fair dice.