The 36 rolls behind every game

Your opponent has a checker on the bar and only the six-point is open, so they need a six to come back in. What are their chances?

One in six is a tempting answer, but either of the two dice can show the six they need. The chance is actually 11 out of 36, a little under one in three.

Looking at all the possible rolls makes the calculation easier to follow.

Give the dice different colours

Take an ivory die and a red die. Each of the six numbers on the ivory die can be paired with any of the six numbers on the red die, giving 36 possible pairs.

The grid below shows all 36 equally likely pairs, with one square for each result of rolling two fair dice.

All 36 possibilities

The ivory die changes down the rows. The red die changes across the columns. Pick an example to see which squares count.

Why 5–3 is twice as likely as 5–5

To roll a five and a three, you can get an ivory five with a red three, or an ivory three with a red five. They take two different squares in the grid, even though you play them as the same roll on the board.

Double five has just one square: both dice have to show five. There isn’t a second way to arrange it.

So a particular mixed roll, such as 5–3, comes up in 2 of the 36 outcomes, or about 5.6%. A particular double, such as 5–5, comes up in 1 of the 36, or about 2.8%.

The 21 rolls often mentioned in backgammon are the 15 mixed rolls and six doubles. They are not equally likely: each mixed roll occupies two squares in the grid, while each double occupies one.

Finding that six

Choosing “At least one 6” lights up a full row and a full column, with six squares in each direction.

Adding the six squares in the row and the six in the column gives twelve, but double six belongs to both. Count it only once and there are 11 squares out of 36, about 30.6%.

The same count works for any number. At least one ace, at least one four, at least one six: all have the same chance. “At least one” includes the double. If you wanted exactly one six, you would leave double six out and count ten squares.

Six ways to make seven

Now choose “A total of 7”. You’ll see 1–6, 2–5, 3–4, 4–3, 5–2 and 6–1. Six squares, so a one-in-six chance.

A total of twelve has only one square, 6–6. The possible totals aren’t equally likely, even though each square is.

Here we mean the numbers showing on the two dice, added together. Backgammon has another wrinkle: you play a double four times, and blocked points can stop you using a roll. A total on the dice isn’t a promise that a checker can move that distance. The grid is the starting point; the position tells you which rolls are useful.

An open point can still be unavailable if entering there would waste the other die. Try both orders of a 6–1.

And the doubles?

They sit neatly along the diagonal: 1–1, 2–2, all the way up to 6–6. That makes six squares out of 36. Any double has a one-in-six chance.

A one-in-six chance does not schedule a double every sixth turn. Fair rolls can put several doubles close together or leave long gaps between them.

You can use that same one-in-six chance in a dice puzzle where two equally likely patterns race to appear first. It is a probability game, rather than a backgammon position. For the app itself, we explain how CUBE Backgammon generates its dice and lets you check recorded rolls.

Use the squares before the percentage

At the board, it is often easier to think “eleven rolls hit me” than to reach for a percentage. Once you can find those eleven squares, the probability follows.

Try a few questions below. They use the same grid as the opening dice lesson in CUBE Backgammon’s Adventure. There is no clock, and you can check every answer.

A note on the counting

These examples assume two fair, independent six-sided dice on an ordinary roll. The opening roll of a backgammon game is different: each player rolls one die, and ties are rerolled. Every fraction here can be checked by counting the matching squares. For a classic treatment, see Oswald Jacoby and John Crawford’s Basic Probability.