Basic Probability
Oswald Jacoby and John R. Crawford, 1970
From The Backgammon Book, Chapter 4 (pp 87–92)

When players are evenly matched, backgammon is essentially a game of probability. Theoretically, you could then take a computer and work out your mathematical chances in any position. Practically, however, there are so many factors involved — including the kind of game your opponent plays — that even the world’s largest and most sophisticated computer would be unable to handle this problem, except perhaps in the very late stages of the game. Experience will teach you that in some situations and against some kinds of players you may be wise to forget about the odds and concentrate on your opponent’s strategy. But in the long run the more you know how to figure the odds, the better the game you’ll play.

A word of warming: don’t expect to assimilate everything in this chapter the first, second, or even third time you read it. Go through it carefully now, and then go on to the next chapter — but get in the habit of referring back to it often as you read the rest of the book. You’ll learn how to figure the odds more and more quickly and easily, and many key principles will eventually become second nature to you.

Now let’s see what there is to learn, without the use of a computer. As a starter here are the thirty-six rolls possible with two dice.

### Table 1Combinations of the Dice

 Double 1 1 Double 2 1 Double 3 1 Double 4 1 Double 5 1 Double 6 1 1 and 2, or 2 and 1 2 1 and 3, or 3 and 1 2 1 and 4, or 4 and 1 2 1 and 5, or 5 and 1 2 1 and 6, or 6 and 1 2 2 and 3, or 3 and 2 2 2 and 4, or 4 and 2 2 2 and 5, or 5 and 2 2 2 and 6, or 6 and 2 2 3 and 4, or 4 and 3 2 3 and 5, or 5 and 3 2 3 and 6, or 6 and 3 2 4 and 5, or 5 and 4 2 4 and 6, or 6 and 4 2 5 and 6, or 6 and 5 2 36

Notice that a double counts as one roll only, while a number such as 6-5 occurs as two dice combinations. Although 6-5 and 5-6 look alike when rolled, they are actually two different rolls. This difference becomes obvious if you are rolling dice of different colors, say one black die and one green die.

Looking a Table 1, you can see that there are eleven ways to roll a specific number, such as 1. Are there, then, twenty-two ways to roll one of two numbers, such as 1 or 2? You might think so, but that isn’t the case, There are only twenty ways, since 1-2 and 2-1 cannot be counted twice.

Similarly, there are twenty-seven ways to roll one of three specified numbers, thirty-two ways to roll one of four, and thirty-five ways to roll one of five.

Table 2 merely reflect the preceding, since the “number of points open” corresponds to the number of different numbers you can roll. Thus if you have a man on the bar and there are five points open in your opponent’s inner board, you want to know the probability of rolling any of five specified numbers.

### Table 2Probability of Entering from the Bar

 Number ofpoints open Ways tocome in Chance ofcoming in Odds in favoror against 5 35 97% 35 to 1 in favor 4 32 89% 8 to 1 in favor 3 27 75% 3 to 1 in favor 2 20 56% 5 to 4 in favor 1 11 31% 25 to 11 against

It is also useful, when you have to leave a blot somewhere, to be able to count the number of rolls that will hit you. When exposing a man to a single shot, the closer you are to your opponent’s threat, the less chance that you will be hit. Thus if exposed to a 1 (an enemy man one point away), there are only eleven ways to be hit, since only a 1 hits you. When you are exposed to a 2, you can be hit in twelve ways; eleven 2s plus double 1.

When a combination or double shot (i.e., more than 6) is necessary to hit you, with one minor exception, the farther away you are the less the chance that you will be hit. It is not difficult to work these chances out by counting, but as a matter of convenience they are listed in Table 3 for all distances from one to twelve.

### Table 3Probability of Hitting a Blot

(This table takes for granted no enemy points between you and the blot.)

 Distanceaway Ways tobe hit Chances ofbeing hit Odds againstbeing hit 1 11 31% 25 to 11 2 12 33% 2 to 1 3 14 39% 11 to 7 4 15 42% 7 to 5 5 15 42% 7 to 5 6 17 47% 19 to 17 7 6 17% 5 to 1 8 6 17% 5 to 1 9 5 14% 31 to 5 10 3 8% 11 to 1 11 2 6% 17 to 1 12 3 8% 11 to 1

You may also be hit by double 4 when sixteen away, by double 5 when fifteen or twenty away, and by double 6 when eighteen or twenty-four away (each of these is, of course, only one out of thirty-six possible shots, or odds of thirty-five to one against. being hit.)

When you hold one or more points between your blot and your opponent’s threatening man, the number of ways you can be hit is reduced. Thus if you hold your two, five, and six points and expose a blot on your bar point to an enemy man on your one point, he can hit your with any 6 or with 4-2 or double 3, but he cannot hit you with 5-1 or double 2, so you are exposed to fourteen rolls instead of seventeen.

When you are exposed to two numbers, the chance that you will be hit is determined by adding the two individual ways together and subtracting duplications. Thus if you are exposed to men six and one points away, you can be hit in twenty-four ways: seventeen ways from the 6, plus eleven ways from the 1, minus the four ways for 6-1 and 5-1, which hit you from either spot. This same total can be arrived at another way: there are eleven direct 6s, plus nine direct 1s that do not include a 6, plus 4-2, 2-4, double 3, and double 2 — still the same twenty-four ways.

You can also check large probabilities of being hit by counting the rolls that miss. Thus, if you are exposed to 6 and 1, the only shots that miss are double 5, double 4, 5-4, 5-3, 5-2, 4-3, and 3-2, for a total of twelve rolls; thirty-six minus twelve gives you the same twenty-four ways.

Make a habit of counting the number of rolls that can hit you when you must expose a blot. There are some unusual figures here. After arriving at the position shown in Diagram 27, you double and proceed to roll a 3-2. You complain a little to the gods that plague backgammon players, and then settle down to make your best play. You can move your blot five, three, two, or no points, but you can’t bring him to safety.

White to play 3-2

After arriving at the position shown in Diagram 27, you double and proceed to roll a 3-2. You complain a little to the gods that plague backgammon players, and then settle down to make your best play. You can move your blot five, three, two or no points, but you can’t bring him to safety.

You don’t have to count to see that you should either move him all the way or not at all; moving him three or two exposes him to far too many possible hits. At first glance you would expect to be in less danger where you are since you are exposed to only one direct shot, but you are better off moving all the way up to your nine point. There are seventeen ways to shoot a 6, plus four ways to shoot 5-2 and 4-3, or twenty-one total ways you can be hit where you are by one of black’s men on your seven or eight point. But there are only twenty ways to hit you with a 2 or 1 if you move up.

Furthermore, there is another reason for you to move up. Once you do so, if you are not hit right away you are going to be home free on your next roll (barring double 1). If you stay where you are, double 3, double 6, or any roll that totals six or less will still leave you in trouble (and there are thirteen such rolls).

Now look at the position shown in Diagram 28. You roll 6-1. You have to move 6 from black’s nine point; your problem is the 1. If you move your blot to black’s ten point, you leave only fourteen rolls instead of fifteen to hit you, but the correct play is to stay exposed on the nine point.

White to play 6-1

The reason is that here you are really worried about the rolls that will allow your opponent to hit your blot and at the same time to cover his own blot in his inner board. If you move up, he can hit you and cover his blot with five rolls (double 3, 3-4, and 3-6). If you stop on the ten point, he can hit you and cover with only four rolls (double 2, double 4, and 4-5).

Let’s see why you are so worried about the hit-and-cover combination. The reason is that if he hits you and fails to cover, you have twenty ways to come right in, and eleven of these twenty ways allow you to come in and hit his blot. If he hits you and covers, you have only eleven ways to come in at all. If you don’t come in immediately, you will be doubled (if it is a gambling game or tournament play) and you would have to concede the game right away. The rules don’t compel you to refuse that double, but if you are playing for anything of value you really can’t afford to take it.

Table 4 lists the probability of bearing off your last one or two men in one or two rolls, in all combinations. Mastering it is essential in order to know how to move your men at the very end of the game, and in considering late double and redoubles:

### Table 4Probability of Getting the LastOne or Two Men Off in One or Two Rolls

 Totalpoints to go Points on whichyour man or menare located In one roll youhave the following: In two rollsyou havethe following: Number ofwinning rolls Probabilityof winning Probabilityof winning 12 6-6 4 11% 78% 11 6-5 6 17% 88% 10 5-56-4 68 17%22% 92%93% 9 5-46-3 1010 28%28% 96%97% 8 4-46-25-3 111314 31%36%39% 98%99%99% 7 6-14-35-2 151719 42%47%53% 99+%99+%99+% 6 3-35-1 or 4-26 172327 47%64%75% 100%100%100% 5 3-24-15 252931 69%81%86% 100%100%100% 4 2-23-1 or 4 2634 72%94% 100%100% 3 2-1 or 3 36 100% 100% 2 1-1 or 2 36 100% 100%

The part of the table showing the chance to get off in one roll is most important. We don’t expect you to memorize it, but we really hope that you will study it and learn how to work out these one-roll chances, for the following reasons:

First, it is crucial that you know how to move your last few men as you are bearing them off. Suppose that you have one man each on your six, five, and two points. You roll 3-2 and use the 2 to bear off your man from the two point. What do you do with the 3? If you move from the five to the two point, you leave yourself with men on the six and two points and a 36 per cent chance to get off on your next roll. If you move from the six to the three point, you leave yourself with men on the five and three points and a thirty-nine per cent chance to get off on your next roll. Clearly, this second way is correct.

Secondly, in a gambling game and in tournament play you must know when to double in situations where your opponent is sure to get off if he gets to play again (i.e., to win, you must get off in your next roll), and when to accept a double by your opponent.

As you can see in the table, when the total count for your two men is eight or more your chance to get off in one roll is always less than even money, and you should never double. When your count is seven, if your men are on the five and two points you can double; otherwise your chance is still less than even money and you should not double. In all cases where your total two-man count is six or less, you have a good double, except when you have two men on the three point.

Looking at the table with your opponent’s prospects in mind, you can see that when he is down to one man on his six point he has exactly a 75 per cent chance to win, and you accept or refuse a double as you choose. At three-to-one odds, you stand to lose just as much, whether you accept or refuse.

When his two-man total is 5 or less, his chance to get off in one roll is better than 75 per cent. You don’t need to memorize this table, but you can and should learn how to count the number of winning and losing rolls in these situations. Suppose, as an example, that your last two men are on your five and two points.

You’ll get both your men off if you roll and 6 or 5 except 6-1 or 5-1. There are twenty ways of rolling a 5 or 6, and since four of them (6-1 and 5-1) don’t get you off, you start with sixteen winning rolls; in addition, double 4, double 3, and double 2 will win for you, so you thus have a total of nineteen winning rolls.

Now let’s check our accuracy by counting your losing rolls. You lose with any ace or with 4-3, 4-2, or 3-2. There are eleven ace rolls, and the others add up to six more, for a total of seventeen losing rolls. Thirty-six less seventeen equals nineteen winners.

The practical value of the last column of Table 4 is that it points out that you are a tremendous favorite to get off in two rolls any time that you are down to just two men in your home board. You should always double if your opponent has three or more men left to bear off, and he should refuse in all cases except one.

That one case is when both your men are on the six point, and your opponent will be able to bear off all three of his men in one roll if he gets any of the five largest doublets. If you wonder why he should accept a double when your chance appears to be 78 per cent (better than three to one), the reason lies in the fact that you may not get that second roll at all. Combining these factors leaves your net chance of winning less than 75 per cent; therefore, accepting your double is a proper gamble by your opponent, since the odds against him are less than three to one.