Backgammon odds and probabilities: the numbers behind every move

Two dice with six faces each give 36 possible rolls. The key detail is that a 6-5 and a 5-6 are two different rolls: picture one red die and one white die, and you'll see that 6-5 can happen two ways. A double such as 6-6 happens only one way. So there are 30 non-double rolls (15 combinations, each counted twice) and 6 doubles.

From there, every probability in backgammon is a fraction of 36. Count the rolls that work, divide by 36, and you have your chance. The chart below gives the counts you'll use most often.

Two numbers are worth learning by heart. 11 rolls out of 36 contain a given number: the 5 rolls where that number appears with another one, counted twice, plus the double. And one roll in six is a double.

A blot is a single checker on a point, and it can be hit by any enemy checker that lands on it. How likely that is depends on one thing above all: the distance between the blot and the checker that aims at it.

Direct shots: 1 to 6 pips away

When the blot is 6 pips away or less, your opponent can hit it with a single die: that's a direct shot. Any roll containing that number hits (11 rolls), and combinations add to it. A blot 6 pips away is hit by any 6, but also by 5-1, 4-2, 3-3 and 2-2: 17 rolls in total.

Indirect shots: 7 pips away or more

Beyond 6 pips, only the two dice together, or a double, can reach the blot: an indirect shot. The numbers fall off a cliff. A blot 7 pips away is hit by 6-1, 5-2 and 4-3 only: 6 rolls out of 36, about one in six. At 11 pips, only 6-5 hits.

Three patterns stand out. Among direct shots, closer is safer: 11 rolls hit from 1 pip away, 17 from 6 pips away. Stepping from 6 to 7 pips nearly divides the risk by three. And beyond 12 pips, only one double can ever hit: distances such as 13 and 14 can't be hit at all by a single checker.

To decide whether a hit or a blot is worth it, see when to hit in backgammon.

You don't need a chart in front of you. Strong players count shots in a few seconds with the same three steps.

1. Count the direct numbers

Give 11 rolls to each number that hits directly. A blot 4 pips away from one enemy checker: any 4, 11 rolls.

2. Add the combinations

Then list the rolls that hit with both dice or with a double, and that you haven't counted yet. For the blot 4 pips away: 3-1 and 1-3 (2 rolls), plus 2-2 and 1-1 (2 rolls). Total: 15 rolls, a little over 41%.

3. Remove what's blocked or counted twice

If the hitter has to pass through a point you hold, a combination may not get through. Say your blot is 8 pips away and you hold the point 4 pips in front of the hitter: 4-4 and 2-2 now land on your point, so only 6-2 and 5-3 hit. 4 rolls instead of 6.

When a blot faces two enemy checkers, or when you leave two blots, add the shots of each and subtract the rolls you've counted twice. A blot exposed to checkers 6 and 1 pips away: 17 rolls plus 11 rolls, minus the 4 rolls that hit from either spot (6-1, 1-6, 5-1, 1-5). That's 24 rolls, two chances in three. Every extra hitter adds shots, and two direct shots add up fast.

A hit checker goes to the bar and must enter through your opponent's home board before you can move anything else (how entering works). Each die gives you one try: a die enters if the matching point isn't made. Fail with both dice and you dance, losing your whole turn.

Two lessons come straight out of the chart. With one checker on the bar, you still come in more often than not even against four points made (20 rolls out of 36). The real wall is the fifth point, then the sixth: against a closed board no roll enters at all.

With two checkers on the bar, both have to come in before you can play anything else, so each die must enter separately. Against just two points made, both enter only 16 rolls out of 36. That's why hitting a second checker, when it's safe enough, is often so strong: it can cost your opponent several turns.

At the end of a race, the question is simple: can I take off my last checkers this turn? The answer depends on where they stand, not only on how many pips are left. Here are your chances of bearing off your last two checkers in one roll.

Look at positions with the same pip count. Two checkers on the 3-point (6 pips) come off with 17 rolls, while a checker on the 4-point and one on the 2-point (also 6 pips) come off with 23. Checkers on the 4- and 1-points (5 pips) come off with 29 rolls; on the 3- and 2-points, with 25. Where the checkers sit changes the odds as much as the pip count does.

That's wastage at work: rolls whose pips can't all be used. It's also why the race rules of thumb ask you to add a few pips for stacked checkers, and why short bear-offs are best judged position by position. For the technique, see how to bear off fast and safely.

Knowing the numbers is only useful if they change your moves. These are the situations where they matter most.

• Choosing where to leave a blot. If you must leave one within direct range, a blot closer to the hitter is usually safer. If you can, leave it 7 pips away or more: going from 17 shots to 6 is a huge saving.
• A blot facing several shooters. One blot facing two enemy checkers can be hit far more often than the same blot facing one. Count both before you decide.
• Building your board. Each extra point you make takes rolls away from your opponent's entering chances, and each one takes more than the last: the fourth, fifth and sixth points you make take away 7, 9 and 11 rolls.
• Attacking. Putting a second checker on the bar changes your opponent's chances of coming in completely. The chart above shows by how much.
• Using duplication. If the number that hits your blot is also the one your opponent needs elsewhere, they often can't use it for both, and you may lose less to it.

The odds are only half of every decision. The other half is what happens next: what you lose if you're hit, and what you gain if you're missed. A play that leaves 17 shots can be right if being missed wins the game; a play that leaves 11 can be wrong if being hit loses a gammon. Strong players weigh both, and that combination is what an engine measures as equity.

You'll find the same trade-offs in the best opening moves, where some of the strongest plays leave a blot on purpose, and in backgammon doubling strategy, where the 25% take point is itself a probability.

Probability also tells you what normal bad luck looks like. A double comes once every six rolls on average, yet the chance of rolling no double in six rolls in a row is about one in three (5/6 multiplied by itself six times: 33.5%). A 6-6 is a 1-in-36 roll, but over 36 rolls the chance of seeing at least one is only about 64%. Streaks are part of the dice, not a sign that something is wrong.

Over one game, luck can decide everything. Over many games, the rolls even out and decisions make the difference. That's why serious players judge their play by the quality of their decisions, not by single results.

You don't need to memorize every chart. A handful of anchors covers most of what comes up in play:

• 36 rolls, 11 per number, 6 doubles. Everything else is built from these.
• 17 shots at 6 pips, 6 shots at 7 and 8 pips. The cliff between direct and indirect shots.
• Against 4 points made, one checker enters 20 times in 36. Against 5, only 11.
• 8⅙ pips per roll on average. Your link between the pip count and the number of rolls left.

Then count during real games. Before you leave a blot, count the shots. Before you hit, look at your opponent's entering numbers. It feels slow for a few games and then becomes automatic. If any term here is new to you, the backgammon glossary explains it, and the backgammon rules cover every mechanic.