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The math needed to improve your backgammon skills

Written by Micky Sean
Wednesday, November 05, 2008

Many players play backgammon strictly for fun and that’s fine but if you truly wish to become a skilled player at backgammon, you need to learn the math. All great players fully understand the probability of the dice.

The Probabilities of the Dice

There are 36 possibilities when rolling two dice. Of these possible combinations, six of them are doubles. Any non-double roll can be rolled two different ways, making that roll twice as likely to happen as a given double. Using two different coloured dice makes it easier to understand this concept easier. So say one dice is white and other black. You can roll double 6’s only one way – a black 6 and a white 6. However, you can roll 11 two ways – a black six and a white five or a white six and a black five.

When adding up the roll of dice, there are a number of rolls that will make that amount. For instance if you need to roll a total of four, you could roll a black 1 – white 3, white 1 – black 3 or a white 2 – black 2 giving a total of three ways to roll a total of 4. Here is a breakdown of all 36 rolls.

Total Number of Ways
2 1
3 2
4 3
5 4
6 5
7 6
8 5
9 4
10 3
11 2
12 1

Also of those 36 rolls, this is the number of rolls that either contain the number needed or a total of the dice or double.

# Needed # of Rolls that have that number or add up to it
1 11
2 12
3 14
4 15
5 15
6 17
7 6
8 6
9 5
10 3
11 2
12 3
15 1
16 1
20 1
24 1

When entering from the bar, depending on how many pips are open will decide how many rolls are usable.

Pips Open Rolls Usable
1 11/36
2 20/36
3 27/36
4 32/36
5 35/36
6 36/36

Here’s a quick rundown on how to use this:

You are at the end of the game, you have checkers on your 5, 4, and 2. Your opponent is down to three checkers on their 1. It’s your roll and you roll 6-1. Obviously, the six is used to take off the 5 checker but what about the one? Does it matter?

If you move the 2 to the one, it leaves the 4-1 combination. So any 4, 5, or 6 on your next roll you will win. The odds of rolling a combo of dice with a 4, 5, or 6 is 27/36. Double 2’s and 3’s you will also win. Giving you 29 rolls that you can win with.

If you move the 4 to the 3, it leaves you with a 3-2 combination. So any roll with a 1 is a losing roll. Rolling a one is 11/36 leaving 25 good rolls. Therefore the 4-1 is the better move.

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