Mathematics of Backgammon
I know that a lot of people will say “I just play backgammon for fun; I don’t want to learn all this mathematical stuff.” But there is simply no way you can be a good backgammon player without some basic understandings of dice probabilities.
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Dice Probabilities
When rolling 2 dice, there are 36 possible combinations. 6 of these are doubles, the other 30 are not. A given non-double roll is twice as likely to come up as a double roll. To understand this, pretend the dice are different colors – say, one red and one green. You can roll a 6-5 with a red 6 and a green 5, or a red 5 and a green 6. But you can only roll 6-6 with a red 6 and a green 6.
Of the 36 possible rolls, the following are the number that will give a given total:
2
1
3
2
4
3
5
4
6
5
7
6
8
5
9
4
10
3
11
2
12
1
Of the 36 possible rolls, the following are the number that will give you some combination of one or more dice:
| 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 |
The odds on rolling a particular number, when the following number of numbers are available, (for example, the odds of entering from the bar with the following number of open pips) are:
| 1 | 11/36 |
| 2 | 20/36 |
| 3 | 27/36 |
| 4 | 32/36 |
| 5 | 35/36 |
| 6 | 36/36 |
Let me give a quick example of how to use all this. Not long ago I was watching a match where a player was down to three checkers, on his 6, 4, and 2 points. His opponent had three checkers on his ace-point. He was on roll, and rolled 6-1.
It was obvious to play 6-off. But where to play the one? Clearly he wants to maximize his chances of getting off on the next roll and winning the game, if his opponent doesn't roll doubles. Which is better, 4 and 1, or 3 and 2?
We can calculate this quickly. 4 and 1 wins anytime you roll a 4, 5, or 6. Since the odds of rolling any of 3 number are 27/36, you have 27 good rolls, plus double 2's and 3's, for 29 good rolls.
3 and 2 wins anytime you don't roll a 1. Since the odds of rolling any specific number are 11/36, you have 11 bad rolls and 25 good ones.
Thus, leaving 4 and 1 is clearly superior - it increases your chances of winning the game by 11%.
Admittedly, this kind of situation will not come up often. But the point of backgammon is that there are hundreds of different situations that arise, and if a good player gets them all right and you don't, you are giving him a huge advantage over time.