Basics

Dice Theory

Tactics

Glossary

Dice Probabilities

Many decisions in a game of backgammon depend on understanding dice probabilities. There are 36 different outcomes when two dice are thrown, all of them equally likely-six ways of throwing the first die, and for each of them, six ways of throwing the second die. Note that rolls of 1-4 and 4-1 are different outcomes, which means that there are two chances in 36 of throwing a 1 on one die and a 4 on the other.

Suppose you are choosing between two different moves, each of which will expose one of your checkers to being hit. One move will expose a checker that's one point away from opposing checker; the other move will expose a checker that's nine points away from an opposing checker. Which move is riskier, and by how much? To answer this question, simply count the number of throws that will allow the opponent to hit you in each situation. When the opponent is one point away from your blot, 11 rolls will allow the opponent to hit you: 1-1, 1-2, 2-1, 1-3, 3-1, 1-4, 4-1, 1-5, 5-1, 1-6, 6-1. That's 11 chances out of 36. If instead your blot is nine points away, the chance of its being hit is only 4 out of 36 (the opponent would need to roll 3-6, 6-3, 4-5, or 5-4).

Doubling Theory

Deciding when to double and what to do if your opponent doubles can be difficult. As a general rule, a player should double with a solid lead--doubling with only a minimal lead is a mistake, except late in the bear-off stage, because the cube is valuable to own--and if your minimal advantage should become a disadvantage, you don't want your opponent to have the option of deciding whether to redouble.

With a good chance to gammon the opponent, though, you should never double, since that would allow the opponent to escape with a loss of one point instead of two.

You should accept a double, in general, when you estimate your winning chances as being at least 25 percent. Here's why: Suppose you play four games, and in each game your opponent doubles when your chances of winning are exactly 25 percent. If you pass each time, you'll lose four points (one for each game). If you accept each time and play the games out, on the average you'll win one and lose three, which means you'll win two points and lose six points at the doubled rate--still a net difference of four points. So 25 percent is the break-even point. The problem with this rule of thumb is that, in practice, it's impossible to calculate the exact odds of winning during the game, except very late in the bear-off stage.

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There are other factors, though, that affect whether to accept a double. Ownership of the cube has value, so if a decision is close, you should probably accept the double, particularly if you think some good rolls could put you in a position to redouble. If you have a chance of being gammoned, however, you need much better odds than 25 percent to accept.

You should be more inclined to double when no one owns the cube, and less inclined when you own it, since in the latter case you're giving something up that was yours. After all, being able to force the opponent to drop out once your chances of winning exceed 75 percent is significantly better than having to play out the game to the end every time.

Another consideration in doubling is the score of a match. If you're trailing 7-1 in a 10-point match and your opponent doubles, you need to compare the chance of winning the match if you pass (falling behind 8-1) with the chance of winning the match if you accept (risking a 9-1 deficit in order to close the gap to 7-3). Such choices must also be influenced by whether either side has gammon possibilities.

When a game has reached the point of being a straightforward race with no further contact between the checkers possible, the pip count--the sum of the points you must roll to bear off all your men--is an important factor in doubling. With around 100 pips left in the race, a rule of thumb is to double when you are at least 10 pips ahead, and to accept a double when you are not more than 13 pips behind. With fewer pips to go, you can double with a slimmer lead, and you need to have a smaller deficit to accept.