Dice Combinations in Backgammon (Easy Hacks)

What's up backgammon fans? This is Marc Olsen from backgammongalaxy.com. This post is going to be about the dice combinations of backgammon and the probabilities of the dice.

Backgammon is a strategy game, but it's also a dice game, a probability game because we're playing with two dice each. So how do we calculate the probabilities over the board? Well, we have two dice. Each has a uniform distribution, a die has six sides, and the same probability for each of the sides. So one each of the numbers has the probability of one in six, one-sixth. 

The same is true with the other die; which means that the probability of rolling a double 1 is 1/6 times 1/6. So the probability that you will roll double aces is 1 out of 36. 

What about the number 2-1, is that also 1 in 36? No, it's not, because in backgammon, whether you roll an ace with the first die and a deuce with a second die or vice versa (a two with the first die and an ace with the second die) doesn’t matter. In other words, it is the same roll, which means that you have 2 out of 36 dice combinations to roll a non-double. So 1 out of 36 to roll a double, and it doesn't matter which double it is, Whether it's double six, double five, double four, double three, double two, or double one, all combinations have the same possibility. And the non-doubles have a probability of 2 out of 36 or 1 in 18.

How do we use this to our advantage? Well, ou can actually calculate the exact winning chances for you in many positions by using the dice combinatorics. So let's count them out here. 

Here is my system of counting dice numbers:

I use my fingers to count. I start out by counting all the non-doubles, and then I save the doubles for the end. I start from the top.

The highest non-double roll is a six and a five. And then I simply count them downwards. That is, I firstly count;

• six-five, 

• six-four, 

• six-three, 

• six-two, 

• and then six-one. 

And that was all the sixes. Then, I go to fives. I start with five-four because I already counted five-six in the previous count when I counted the sixes. So then I go to; 

• five-four, 

• five-three, 

• five-two, 

• five-one. 

After that, I will go to; 

• four-three, 

• four-two, 

• four-one. 

Then, three-two and three-one. And lastly, two-one.

After that, I count the doubles at the end, and then I use my fingers. In my counting system, each full finger represents 2 out of 36, which would be a non-double. Half a finger represents 1 out of 36, which would be the doubles. 

Doubles: Half a finger

Non-doubles: Full finger

So let's look at the position above and try to count it out using my finger system here. So I start from the top. So I'm going to count six-five; that's one finger. Then six-four, six-three, and six-two.

In this position, six-five, four-three, and two work. Let's go to the fives. Five-four, five-three also works, and lastly I go to five-two. 

What about the fours? Well, we already counted six-four and five-four, so we have to go to four-three but in this case four-three doesn't work since it doesn't bear off both checkers. 

Okay, so we have seven fingers coming from the non-doubles. That's 14 out of 36 because, remember, each finger represents 2 out of 36. Now, I'm going to count the doubles: double six, double five, double four, double three, and double two, leading to nine and a half fingers, which is 19 out of 36 dice combinations for white to win this game. So white is a favorite. In this case, if white has access to the cube, he should most definitely double since it's the last roll, and the cube cannot be recubed. So if you have the slightest edge, you should cube, and because you're a 19 out of 36 favorite in this position. You can see that calculating the dice combinations helps you build strategies for the positions you encounter.

Example Positions for Different Dice Combinations

Let's have a look at another position where we use dice statistics to our knowledge. Imagine that you, unfortunately, just rolled a 6-1, so you have to leave a shot. The six is forced; we take out the six and the ace. 

You cannot play it safe, so you have to expose yourself and leave a shot. You should now think: What is the probability of green hitting the white checker? Well, again, let's count. So now we're going to count all of the dice combinations that include an ace. We have 6-1, 5-1, 4-1, 3-1, 2-1, which constitute all the non-double combinations. These combinations add up to five fingers, a possibility of 10 rolls out of 36. And now we need to count the doubles: double one. That's the only one, so that's half a finger. Adding it to the possibility of 10 rolls out of 36 coming from non-doubles, it gives us a result of 11 out of 36 to hit a direct shot. That's a good thing to remember every time you have a direct shot. If you don't have any additional numbers that hit, it's always 11 out of 36.

What happens if we take the unsafe checker one pip back, put it here instead?:

Now it's a deuce that hits instead of an ace. So again, all dice combinations that include a deuce with two dice have a probability of 11 out of 36. However, here there is an additional hit; green can also roll double one. So you have to add the additionals to the direct shot. So we have 11 out of 36 that hits with a deuce directly, and you have double aces that also hit the checker: a total of 12 out of 36.

What if we put it one more pip back? 

Well, again, we have all the threes, which are 11 out of 36, and now we have to count the additionals. Here we have two-one and double one, so that's one finger and a half, three extra numbers, which is a total of 14 out of 36. 

We'll move it one pip back, the direct four, which again leads to a possibility of 11 rolls out of 66. 

Let's count the additionals: three-one, double one, and double two. That's two full fingers, and four extra shots. In this case, you get a possibility of 15 rolls out of 36 that hit the checker when it's four pips away. What about five pips away? 

Again, 11 out of 36 with the fives. Counting the additionals: four-one, three-two, and three-one. That's it; no doubles hit this checker. So again, we have 15 out of 36. 

And now we slide it over to six pips away. 

Again, 11 out of 36 to hit with a direct six. Let's count the additionals: five-one, four-two, double three, and double two. So that's three full fingers, six extra numbers. 17 out of 36 to hit the direct six.

What about if we take it one more pip, so it's now seven pips away? 

Now it's out of the direct range and there's no direct shot, so now we can't apply that rule anymore. We have to count it again. Which numbers hit seven pips away? Six-one, five-two, four-three. So, the probability of green hitting the white checker is 6 out of 36.

In conclusion, all that I explained above was basically the fundamentals of dice probabilities in backgammon. I hope you can apply it to your advantage over the board. I hope to see you on Backgammon Galaxy or maybe at a live tournament where you are playing on the Earth Board designed by Galaxy!

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