# Possible Outcomes of Rolling Two Dice

Since all backgammon games are played with two dices, it is interesting to examine the possible outcome of rolling two dice.

## One die

As we know, the die is a cube with numbers printed on each side. The numbers that are printed are 1, 2, 3, 4, 5, 6, and are not printed randomly. Instead, the sum of two opposite sides is equal to 7. Thus, 1 is opposite to 6, 2 is opposite to 5, and 4 is opposite to 3.

When we roll an unbiased die, the probability to roll one of these 6 numbers is 1 / 6. All numbers have that same probability. Things, however, get more interesting when we roll two dice.

## Two dice

When we roll two dice the result can be grouped into two categories: Both dice rolled the same number and the numbers are different. Even though this looks trivial, the mathematics that govern each result are different. Since we have two dice, the total combinations from rolling them is 36.

### Probability for rolling the same number

This is know as doubles, and it means that both dice rolled 1 or 2 or … The probability to get one of these result is 1 / 36. Because there are 6 numbers, the probability to get any of these results is 6 / 36 = 1 / 6.

### Probability for rolling different numbers

This means that one die rolled 1 and the other rolled 2 or 3 or 4 or 5 or 6, but not 1. Now, in mathematics the result 1 – 2 is different from the result 2 – 1. But, in a backgammon game they are exactly the same. Thus, the probability to roll a specific combination is 2 / 36 = 1/ 18.

The above analysis leads to the following conclusions:

• Rolling a specific double is highly unlikely
• Rolling any double is more likely than rolling a specific combination

## Probability for rolling a certain number

Up until now we examined each die separately, Now, we sum the outcome of each die, and we examine the probability of rolling a certain number. This sum ranges between 2 and 12. A sum equal to 2 or 12 is the least likely to get (e.g., 1 / 11). On the other hand, a sum equal to 7 has the highest probability (e.g., 6 / 11). Following, a sum equal to 6 or 8 are the second most likely to obtain (e.g., 5 / 11).

## Probability to move a given distance

Now that we learned a lot about the probability of getting a certain number, how can we protect our checkers? In a backgammon game a checker can be moved between 1 and 24 spaces. Note that that range is twice as big as the range of summing the outcome of two dice. Also, in a backgammon game we can move a certain distance either because the sum is equal to that distance or because one die is equal to it. Thus moving a checker by 6 spaces has the highest probability (10 / 24). The second highest probability has to move a checker by 4 spaces (9 / 24), and the 3rd highest probability to move it by 5 spaces (8 / 24).

## Conclusion

We did not want to bother you with rigorous mathematical definitions and calculations. Thus, we gave the computed probability of the outcome of rolling two dices. We examined the cases of rolling a double and moving a checker in a given distance. We hope that we helped you to understand the complexity of playing a backgammon game.