Backgammon odds and probabilities

Backgammon is a game of dice, but the dice follow simple rules. Every number on this page comes from listing all 36 rolls and counting, so you can check each one yourself.

The 36 rolls

Two dice with six faces each give 6 × 6 = 36 equally likely outcomes. Treat the dice as different (imagine one red and one white) and the counting becomes easy: a non-double like 6-5 can come up two ways, 6 on the red and 5 on the white or the other way round, so it has 2 chances in 36. A double like 6-6 can only come up one way, so it has 1 chance in 36.

That gives 15 non-doubles × 2 = 30 outcomes, plus 6 doubles × 1 = 6 outcomes, for 36 in all. Remember that a double is played four times: 4-4 moves 16 pips. That is why doubles matter so much in races and in shot counts, and why the average roll is worth 8.17 pips rather than 7.

Why 11/36 and 17/36

The most useful single fact in backgammon odds: the chance that a given number appears on at least one die is 11/36, about 30.6%. There are 6 rolls with that number on the first die and 6 with it on the second, but the double (say 6-6) is in both groups, so 6 + 6 - 1 = 11.

Now look at a blot 6 pips away. Every roll containing a 6 hits it directly: that is the 11. But other rolls hit it by combining both dice: 5-1 and 4-2 (two ways each) and the doubles 3-3 and 2-2 (one way each). That adds 6 more, for 17/36, about 47.2%. This is why a blot 6 pips away is the most exposed spot within direct range. A blot 7 or more pips away can only be hit with combinations, so the count drops sharply: just 6 rolls hit from 7 pips away.

So the quick rule: a blot within 6 pips can be hit by roughly a third to a half of all rolls; a blot 7 to 12 pips away is hit by about one roll in six or fewer; beyond 12 the risk is tiny.

Hitting a blot

The table counts the rolls that hit a single blot at each distance when nothing blocks the way. "Direct" rolls have a die equal to the distance. "Combination" rolls hit only by moving one checker with both dice, or with two, three or four steps of a double.

No roll hits a blot exactly 13, 14, 17, 19, 21, 22, 23 pips away with one checker. The single most dangerous distance is 6 pips (17/36). Beyond 6, notice the small bump at 12 (3 rolls: 6-6, 4-4 and 3-3) compared with 11 (2 rolls: only 6-5).

When points are in the way

Real positions are rarely open. A combination shot needs a landing spot for the first part of the move: to hit with 5-1 from 6 pips away, the checker must be able to stop 5 or 1 pip along. If your opponent has made those points, that roll no longer hits. Blocking points can only take rolls away, so treat the table as the maximum. And when your opponent has more than one checker that can hit, or you leave two blots, count the rolls that hit at least once without counting any roll twice.

Entering from the bar

A checker on the bar enters on the point matching one of your dice, counted from the far end of your opponent's home board. Each closed point (two or more opposing checkers) is a number that does not work. With k points closed, the chance that both dice show a closed number is (k/6)², so the chance to enter is 1 - (k/6)².

The pattern is worth remembering: a 4-point board still lets you in 55.6% of the time, a 5-point board only 30.6%, and a closed board stops you completely until a point opens. When two checkers are on the bar, both must enter before you can move anything else, which needs both dice to work; the last column counts that. Against a 5-point board you get both in only with the one double of the open number.

This is why building your home board before hitting is so strong, a theme covered in strategy.

Other useful odds

Two consequences come up all the time. First, a player who needs one particular number on the next roll is an underdog, roughly 11 to 25 against. Second, needing either of two numbers (20/36) is better than even. When you have a choice of where to leave a blot, or which points to make, the best play often gives your opponent the fewest good numbers rather than none at all.

The average roll of 8.17 pips is the basis of racing arithmetic: being on roll is worth roughly half a roll, about 4 pips. See pip counting for how that feeds into race decisions.

Using the numbers at the board

You do not need the whole table memorised. A handful of facts covers most decisions:

• A number on at least one die: 11/36. Use it as your basic building block.
• A blot 6 pips away: 17/36. Within 6 pips, the closer blots are hit less often, from 11/36 at 1 pip up to 17/36 at 6.
• A blot 7 to 12 pips away: 2 to 6 rolls. Distant blots are much safer.
• Entering against k closed points: 36 - k² rolls.
• Doubles: 1 in 6.

When you count shots in a real position, list each distance from each enemy checker that can reach your blot, take the rolls that hit, remove any that are blocked, and remove duplicates. It sounds slow but becomes quick with practice. Odds also matter for the cube: the doubling cube guide explains why you can take a double when you win at least about a quarter of the time.

Want to see the numbers in action? Play a game and count your opponent's shots before you leave a blot. Use the hint button to check whether the computer agrees with your choice.

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