Backgammon Probabilities

Equal parts luck and skill, the game of backgammon is attractive to all kinds of people. In other words, it’s possible for a newcomer to even beat a pro if he/she gets lucky with the dice. Such precariousness can’t be found in other games such as chess or checkers. Of course, an experienced player will nearly always emerge as the victor in any series of games, but regardless, the luck factor is what draws many new players to backgammon.

Accordingly, in order to improve as a player, one must understand this luck factor. By comprehending the numbers involved and knowing the correct chances of a certain number coming up, you place yourself in a better position to make the correct decisions necessary for victory.

Your checkers are often exposed to the possibility of being hit. A player who understands the probability, and can account for the degree of risk, is in a better position to decide when and where to expose such pieces.

Odds” (a.k.a. ‘probability’) are defined as the ratio of probability that something will occur. Put differently, at some point in a game a player might discover that a pair of 5s is exactly what is needed to execute some plan, or capitalize on an opponent’s vulnerability. The odds of rolling a pair of fives are calculated as follows:

    There is a 1 in 6 chance of a rolled die of showing any one digit. So, in order for any two particular digits to be rolled, you must multiply the odds of each event occurring. For example, 1/6 X 1/6 = 1/36, or 35 to 1.

As you can see, the chances of rolling double-5’s are not very high and, consequently, you shouldn’t count on this occurring.

Possible Outcomes

A die has 6 sides and so, 6 possible outcomes, ranging from 1 to 6. When rolling two dice, you’ll have 36 possible outcomes, (6 X 6 = 36) which are referred to as dice combinations.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

As you can see from the 36 dice combinations, the chances of rolling any particular doublet (for example double-fives) are the same; 1 in 36, or 35 to 1. (NOTE: The combinations in blue are the desired outcomes).

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

What if, however, you need two different numbers, for example a 5 and a 2? Have a look at the table below to determine your odds for rolling this:

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Now, there are two possible dice combinations with which you can achieve your goal. Your odds of hitting this combination have thus doubled; 2/36, or 17 to 1.

Along the same lines, consider the following: at an important stage of the game, you find yourself exactly 7 pips away from a crucial blot, with none of your opponent’s checkers separating your men and the exposed blot. What are the odds of rolling the necessary digits now?

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Here you can see the odds have risen substantially in your favor, compared to the previous two scenarios. You now have a 6/36 (5 to 1) chance of rolling a seven.

Now, consider a situation in which you need only one particular digit to execute a plan, for example a one. Examine the table to determine the odds.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Judging from the table, we see that your chances of rolling a one (or any particular digit) are 11/36, or 30.6%

The final scenario to consider occurs when you are within 6 pips of an important position, and how you roll the dice plays no role in how you get to that position, whether it be a combination or a single digit. For example, one of your checkers is 5 pips from a crucial point (or blot). Look at the table to determine the odds of rolling the needed number.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

So, as you can see here, the odds of rolling the numbers necessary to move one of your checkers 5 pips are 15/36, or 41.7%