How to...Part 3

Backgammon is a game of probabilities, and, the more you know about them the better your game will become. A lot of players that lose at backgammon complain that the dice were against them! This isn't so, the dice are totally inanimate and have no bias towards either player - it is just the player's conception that they favour their opponent. It is essential if you want to become a good backgammon player that you study and learn (if possible) as much as you can of the following. It is this knowledge that will give you the edge over an opponent and will help you win more games and matches. Your opponent will think you are 'lucky' and may never come to realise that your 'luck' is based upon your superior knowledge of dice probabilities.

You don’t need to be a good at maths to understand the probabilities in backgammon; all you need to be able to do is add up to thirty-six at the most! If you can’t add up to thirty-six then take my advice - don’t take up backgammon!

Dice Combinations

Detailed below are the thirty-six dice combinations. You don't need to remember what the rolls are just how many there are. Just remember that each roll, (2-1, 5-4, 3-2, etc.) is made up of two dice and either die can be the 2 or the 1. This way we get thirty opening rolls, plus six doubles. Diagram 7 illustrates the combinations.

Rolls That Contain a . . .

Diagram 8 below shows all the dice rolls and what numbers they contain. This is a very useful chart and understanding the chances of certain numbers being rolled put the dice in your favour. It’s very important that you understand just how many dice rolls will contain a certain number. For example, do you know how many dice rolls contain a 1? You could look at the dice in the Diagram 7 above and count them, but here’s a shortcut; it’s eleven. Eleven is a base number for all rolls that contain n. For rolls that contain a 2, we start with the base of eleven and add to it all dice rolls that add up to 2 but don’t contain a 2, in this case, 1-1 (double one); so there are twelve rolls that contain a 2. For dice rolls that contain a 4 we start with eleven and add 2-2, 1-1, 3-1 and 1-3 to give us fifteen.

Clearly, direct rolls (rolls that are within the range of one die: 1 to 6) are the easiest to roll with 6 being the most likely one and 1 being the least likely. So, when moving your checkers around the board, take care to minimise your chances of being hit by remembering this chart; or maximise your chances of rolling a certain number. It is very handy knowing that when your opponent leaves a blot n points away there are x number of rolls that can hit them. Again, they'll think you were lucky when in actual fact the odds were probably in your favour.

% chance of rolling a specific number
1 = 30.55%	7 = 16.66%	15 = 2.77%
2 = 33.33%	8 = 16.66%	16 = 2.27%
3 = 38.88%	9 = 13.88%	18 = 2.27%
4 = 41.66%	10 = 8.33%	20 = 2.27%
5 = 41.66%	11 = 5.55%	24 = 2.27%
6 = 47.22%	12 = 8.33%

Diagram 9

Rolls That Hit A Double Shot

Diagram 10 shows how dangerous a checker or checkers are exposed to a double hit, i.e., within the range of 1 to 6 pips away. As you can see in the Diagram 10 below, as many as 20 (55.56%) minimum and going up to 28 (a whacking great 77.78%) dice rolls can hit when exposed to a double hit - something to be very wary of; and of course, the more checkers you are exposed to the greater the odds of being hit.

There's a little maxim often used by backgammon players, "nearest, safest" which means that the closer a blot is to a threatening opponent the safer it is if within direct range and "furthest, hardest" when exposed to an indirect shot (although 12 away is worse than 11 away!).

Rolls that hit a double shot
Numbers that hit	How many dice rolls hit and %

6 &1	24	66.67%
6 & 2	24	66.67%
6 & 3	28	77.78%
6 & 4	27	75.00%
6 & 5	28	77.78%
5 & 1	22	61.11%
5 & 2	23	63.89%
5 & 3	25	69.44%
5 & 4	26	72.22%
4 & 2	23	63.89%
4 & 1	21	58.33%
4 & 3	24	66.67%
3 & 1	20	55.56%
3 & 2	21	58.33%
2 & 1	20	55.56%

Diagram 10

Being hit isn't the only use for double shots; the same odds are applied to making a point on your next roll. If you're not in too much danger of being hit by your opponent's next roll then the more chances you have of making a point the better.

To recap; the base number for rolling a single shot is eleven, the base for a double shot is twenty . . . and the base for a triple shot is twenty-seven. Just remember the bases, you can add the ‘extras’ when you need them.

Re-entering Off The Bar With One or Two Checkers

We began with bearing off, and within we dealt with being hit and re-entering off the bar. Of course, if you are hit - after being exposed to a double hit as above, for instance, - it’s as well to know what the odds are of re-entering. The inability to enter off the bar can often lead to losing the game and therefore it is advisable to know what rolls will enter and what won’t. Diagram 11 shows the odds of re-entering with one or two checkers against n points closed.

Re-entering off the bar with 1 or 2 checkers
Closed points	Number of entering rolls	One checker	Two Checkers
1	35	97.22%	69.44%
2	32	88.88%	44.44%
3	27	75.00%	25.00%
4	20	55.56%	11.11%
5	11	30.56%	2.78%

Diagram 11

Quite obviously the more points closed the harder it is to re-enter . . . but not as hard as most beginners think! For example, often new players reckon that with one checker on the bar, if three points are closed (50%) then only eighteen rolls (50%) will re-enter; this is incorrect. Take a look at the table showing three closed points - a total of twenty-seven rolls re-enter, that's 75%, nowhere near the expected 50%. In fact with as many as four closed points you're still favourite to re-enter one checker with twenty rolls (55.56%). But, with two checkers on the bar these odds change dramatically as.

Bearing Off Your Last Two Checkers

We began this tutorial at the end, Bearing Off. So, our final table is all about the odds of getting your last two checkers off in one roll. This is a knowledge that too few players ever bother to attain; but one which can mean the difference between winning or losing. Too many backgammon players don’t know on what points to leave their last two checkers to have the best chance of bearing them off with one dice roll. Obviously the lower you leave them the more rolls will do the job. Without referring to the Diagram 12 below, which of these two points do you think would be the better, 3 & 4, or 2 & 5?

Bearing off your last two checkers
Checkers on points ...	Off in n rolls and %
1 & 3	34	94.44%
1 & 4	29	80.55%
1 & 5	23	63.89%
1 & 6	15	41.47%
2	26	72.22%
2 & 3	25	69.44%
2 & 4	23	63.89%
2 & 5	19	57.78%
2 & 6	13	36.11%
3	17	47.22%
3 & 4	17	47.22%
3 & 5	14	38.89%
3 & 6	10	27.78%
4	11	30.56%
4 & 5	10	27.78%
4 & 6	8	22.22%
5	6	16.67%
5 & 6	6	16.67%
6	4	11.11%

Diagram 12

Almost every beginner will go for 3 & 4, and they’d be wrong! The better one is 2 & 5, nineteen rolls vs. seventeen rolls; and in a game where even the slightest advantage can mean winning rather than losing, those two extra rolls are vital. Always try to end up with at least one checker on a 1 or 2, and the second as close to it as possible. Apart from 2 & 6 this is very good advice.

I don't suggest you commit these entire tables to memory but if you can just remember half of them your game will improve. This is basic information that is essential to understanding the probabilities of dice rolls and their repercussions. Armed with the knowledge gleaned from Part III you will have a tremendous advantage over an opponent that hasn’t bothered to learn them. When you start making more points or hitting more blots than they do they’ll think you’re just lucky – let them believe that; whilst they attribute your skill to luck you’ll always have that winning edge over them.

Finally. During the Mary Rose Trophy in February, 1996, Paul Money and Simon K Jones were battling it out for the trophy. At double match point (DMP - a stage in a match where each player is 1 point away from victory) Simon was on roll and he had to bear his last two remaining checkers off on his next roll to win if Paul failed to roll double-two or better on his. If he failed to take them both off, Paul would be the winner. Simon had already played half his roll and he had a 1 left to play in this position:

Simon (black) to move a 1

Simon thought it about for a while, weighing up the chances of getting both checkers off next roll and eventually played 6/5 giving him 19 winning rolls with his checkers on his 5- and 2-points. He rolled 6/1 and lost! If he’d played 2/1 and given himself four rolls fewer he’d have won the title. You might think from this that it pays to be ignorant of the probabilities, but in the long term you will win more games.

So, we're now at the end of this tutorial. I hope I have given you a taste for backgammon and that you're game will be that bit better than an opponent who hasn't bothered to learn the basics herein.

Essential Tables For Winning At Backgammon

Dice Combinations

Rolls That Contain a . . .

Rolls That Hit A Double Shot

Re-entering Off The Bar With One or Two Checkers

Bearing Off Your Last Two Checkers