# How to Calculate Backgammon Probabilities

Backgammon is a game that employs the use of two standard dice.  The dice used in this game are six-sided cubes, and the faces of a die have one, two, three, four, five or six pips. During a turn in backgammon a player may move his or her checkers or draughts according to the numbers shown on the dice. The numbers rolled can be split between two checkers, or they can be totaled and used for a single checker. For example, when a 4 and a 5 are rolled, a player has two options: he may move one checker four spaces and another one five spaces, or one checker can be moved a total of nine spaces.

To formulate strategies in backgammon it is helpful to know some basic probabilities. Since a player can use one or two dice to move a particular checker, any calculation of probabilities will keep this in mind. For our backgammon probabilities, we will answer the question, “When we roll two dice, what is the probability of rolling the number n as either a sum of two dice, or on at least one of the two dice?”

## Calculation of the Probabilities

For a single die that is not loaded, each side is equally likely to land face up. A single die forms a uniform sample space. There are a total of six outcomes, corresponding to each of the integers from 1 to 6. Thus each number has a probability of 1/6 of occurring.

When we roll two dice, each die is independent of the other. If we keep track of the order of what number occurs on each of the dice, then there are a total of 6 x 6 = 36 equally likely outcomes. Thus 36 is the denominator for all of our probabilities and any particular outcome of two dice has a probability of 1/36.

## Rolling At Least One of a Number

The probability of rolling two dice and getting at least one of a number from 1 to 6 is straightforward to calculate. If we wish to determine the probability of rolling at least one 2 with two dice, we need to know how many of the 36 possible outcomes include at least one 2. The ways of doing this are:

(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6)

Thus there are 11 ways to roll at least one 2 with two dice, and the probability of rolling at least one 2 with two dice is 11/36.

There is nothing special about 2 in the preceding discussion. For any given number n from 1 to 6:

• There are five ways to roll exactly one of that number on the first die.
• There are five ways to roll exactly one of that number on the second die.
• There is one way to roll that number on both dice.

Therefore there are 11 ways to roll at least one n from 1 to 6 using two dice. The probability of this occurring is 11/36.

## Rolling a Particular Sum

Any number from two to 12 can be obtained as the sum of two dice. The probabilities for two dice are slightly more difficult to calculate. Since there are different ways to reach these sums, they do not form a uniform sample space. For instance, there are three ways to roll a sum of four: (1, 3), (2, 2), (3, 1), but only two ways to roll a sum of 11: (5, 6), (6, 5).

The probability of rolling a sum of a particular number is as follows:

• The probability of rolling a sum of two is 1/36.
• The probability of rolling a sum of three is 2/36.
• The probability of rolling a sum of four is 3/36.
• The probability of rolling a sum of five is 4/36.
• The probability of rolling a sum of six is 5/36.
• The probability of rolling a sum of seven is 6/36.
• The probability of rolling a sum of eight is 5/36.
• The probability of rolling a sum of nine is 4/36.
• The probability of rolling a sum of ten is 3/36.
• The probability of rolling a sum of eleven is 2/36.
• The probability of rolling a sum of twelve is 1/36.

## Backgammon Probabilities

At long last we have everything we need to calculate probabilities for backgammon. Rolling at least one of a number is mutually exclusive from rolling this number as a sum of two dice. Thus we can use the addition rule to add the probabilities together for obtaining any number from 2 to 6.

For example, the probability of rolling at least one 6 out of two dice is 11/36. Rolling a 6 as a sum of two dice is 5/36. The probability of rolling at least one 6 or rolling a six as a sum of two dice is 11/36 + 5/36 = 16/36. Other probabilities can be calculated in a similar manner.

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