Backgammon Probabilities

1. The 36 dice combinations

Two six-sided dice produce 36 possible combinations (6 × 6). Of these, 6 are doubles (the same number on both dice) and 30 are non-doubles. Here is the basic table:

Odds of individual rolls

Each specific double (e.g. 4-4) has a probability of 1/36 ≈ 2.78%. Each specific non-double (e.g. 3-5, which can come up as 3+5 or 5+3) has a probability of 2/36 ≈ 5.56%.

2. The odds of reaching a point (direct and indirect shots)

A fundamental concept is how many ways you can reach a particular point from a given checker.

Direct shots (1 to 6)

To reach a point N pips away (N ≤ 6) with a single die:

• Distance 1: 11/36 rolls (every roll containing a 1: 1-1, plus 1-2 to 1-6 in both orders)
• Distance 2: 12/36
• Distance 3: 14/36
• Distance 4: 15/36
• Distance 5: 15/36
• Distance 6: 17/36

3. Pip counting

The pip count is the total number of pips your checkers still have to travel before they are all borne off. At the start of a game, each player has exactly 167 pips. Here is how to work it out:

Multiply the number of checkers on each point by that point's number (from your perspective):

• 2 checkers × 24 = 48
• 5 checkers × 13 = 65
• 3 checkers × 8 = 24
• 5 checkers × 6 = 30
• Total = 48 + 65 + 24 + 30 = 167 pips ✓

Interpreting the pip count

• Your pip count < your opponent's: you are ahead in the race
• Your pip count > your opponent's: you are behind
• Difference > 10%: the player in front should consider a pure racing strategy

Quick estimation methods

Working out the exact pip count in the middle of a game is hard. Experts use a few shortcuts:

• The anchor method: estimate the value of the largest groups of checkers
• Simplified pip count: count only the most advanced and the furthest-back checkers
• Rule of thumb: if your runners are still far back while your opponent is moving forward, you are probably behind

4. Race calculations

Once both players' checkers have passed each other (no more contact is possible), the game becomes a pure race. Here is how to estimate your chances:

The Effective Pip Count (EPC)

For a race with no contact, the rule of thumb is: if your EPC (your pip count adjusted to account for “wastage” on your rolls) is lower than your opponent's, you are the favourite.

The Keith count

The Keith count is a practical rule for making cube decisions in a race:

This formula gives a quick approximation but is only valid in reasonably balanced races.

5. Gammon probabilities

A gammon (the loser has not borne off a single checker) happens more often than you might think. Here are a few figures:

• In a normal game between experienced players, around 20–25% of games end in a gammon
• If you still have checkers on the bar or far back when your opponent starts bearing off, the risk of a gammon rises to 40–60%
• These probabilities directly affect your cube decisions

6. Putting it into practice

During the game

You do not need to calculate everything in real time. Here is what a good player assesses quickly:

1. Am I ahead or behind? (a quick estimate of the pip count)
2. Are my blots in danger? (the number of opposing rolls that hit)
3. How likely is my plan to work? (the good rolls I need)

Probabilistic intuition

After thousands of games, experienced players develop a “feel for the odds” — they sense instinctively whether a position is favourable or not, without precise calculation. This comes from regular practice and from analysing your own mistakes.