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The 36 rolls: a probability table for backgammon

Two dice give 36 possible rolls. Almost every decision in backgammon comes down to how many of those 36 help you, and how many help your opponent.

Here are the numbers worth knowing, in one place.

The basics

Question Answer
Rolls containing a specific number (say a 4) 11 of 36, about 31%
Rolls that are doubles 6 of 36, about 17%
A particular double, say 5-5 1 of 36, under 3%
Average pips moved per roll 8.17
Pips each player must travel from the start 167

The first line is the one that can feel surprising. A specific number isn't one in six; it's 11 in 36, because it can appear on either die.

The average roll is higher than you'd guess from two dice averaging 3.5 each, because doubles move four numbers instead of two.

How likely is a hit?

This is the table to memorise if you memorise one. It shows how many of the 36 rolls hit a lone checker at each distance, with no blocking points in the way.

Distance Rolls that hit Chance
1 pip 11 of 36 31%
2 pips 12 of 36 33%
3 pips 14 of 36 39%
4 pips 15 of 36 42%
5 pips 15 of 36 42%
6 pips 17 of 36 47%
7 pips 6 of 36 17%
8 pips 6 of 36 17%
9 pips 5 of 36 14%
10 pips 3 of 36 8%
11 pips 2 of 36 6%
12 pips 3 of 36 8%

Two things fall out of it. Six away is the most dangerous distance on the board, hit by nearly half of all rolls. And the drop from six to seven is enormous: 17 down to 6. One extra pip of distance cuts your risk by two thirds.

Distances of six or less are direct shots, reachable with one die. Seven and over are indirect, needing both dice to cooperate.

Coming in off the bar

If your checker is on the bar, you enter on a point matching one of your dice, so what matters is how many of your opponent's six home points are open.

 

Points open Rolls that enter Chance
6 36 of 36 certain
5 35 of 36 97%
4 32 of 36 89%
3 27 of 36 75%
2 20 of 36 56%
1 11 of 36 31%
0 none closed out, no move at all

This is why making points in your home board matters so much, and why a closed board is worth building even at some cost.

How to use all this

You don't need the whole table at the table. Three facts cover most decisions:

The rest is useful when you're reviewing a game afterwards and want to know whether a risk was worth taking. Safe or bold turns these numbers into decisions.

The short version

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Angela

One of the top 10 sweetest people, probably. German by birth, delightfully unserious by nature. By day, she shapes stories; by night, she beats her boyfriend at backgammon and writes poetry.