# Understanding the Probability and Odds in Backgammon

It’s important for beginners to understand that backgammon is a game that requires both luck and skill. Even with modern technology, it’s impossible to determine the exact percentage of luck versus skill in most gambling games, and backgammon is no exception. In fact, beginners have been known to beat pros with the right rolls of the dice.

However, it’s also true that a single roll of the dice can completely change the course of a game, adding to its excitement and unpredictability. If two players of equal skill are playing, the player with the better dice will likely win. On the other hand, if two players have equal luck, the one with more skill will likely come out on top.

Given the importance of luck in backgammon, beginners should familiarize themselves with the odds. “Odds” refers to the probability of something occurring, or the likelihood of one event happening compared to another.

For example, during a game, you may need double sixes to execute your strategy. But keep in mind that the odds against rolling double sixes are thirty-five to one, which is not a reliable bet.

While it’s important to be aware of the odds, it’s also important to remember that rules are meant to be broken. Sometimes, the odds won’t apply if you’ve learned your opponent’s playing style or if you’re faced with a unique situation.

Here are some odds to keep in mind when playing backgammon. Note that they only apply to backgammon and not other dice games. The odds listed below show the probability of rolling specific numbers or combinations of numbers on the dice.

The following are the odds of rolling the necessary combinations to bring your man off the bar and back into the game in backgammon.

Backgammon Odds: Additionally, it’s important to consider the “feel” of the dice in all games of chance. If you’ve ever played craps, you’ll know the feeling of anticipating your number or a seven coming up.

Improving your skill in backgammon is a logical way to increase your chances of winning. With dedication and practice, your skill will eventually surpass your luck in the long run.

## Understanding Probability in Backgammon

To succeed at backgammon, it is essential to have a grasp of probability – the likelihood of certain events occurring. This principle is not limited to backgammon, but is also the foundation of many business activities, such as insurance. Insurance companies calculate the probability of an event happening rather than relying on luck, and backgammon players must do the same.

Relying solely on luck can be detrimental to a player’s success. While it may help in the short term, luck tends to even out over a series of games. A novice may win against an expert player through sheer luck, but in the long run, the expert’s skill and understanding of probability will shine through.

Even when luck plays a significant role, skill is still necessary to make the most of it. Probability is a scientific method that can aid players in making informed decisions. By understanding the odds of a particular move succeeding, such as covering a blot or avoiding being hit, players can increase their chances of success. Therefore, probability is a valuable tool in backgammon and must be utilized effectively to succeed.

## 36 combinations

A die has 6 sides, and the probability of rolling each number is 1 in 6. If you roll the die 6 times, each number will turn up approximately the same number of times due to the law of large numbers. Relying on lucky rolls may lead to winning a game or two, but over a thousand games, playing according to the laws of probability is the key to success.

In backgammon, there are two dice, and each die has 6 possible outcomes, resulting in a total of 36 possible combinations. This is because each die can be thrown in 6 different ways, and when multiplied together, the result is 36. To illustrate this, consider the following table that shows all the possible combinations when rolling two dice.

### Observations from the Table:

• The probability of rolling a specific double is 1 in 36.
• All non-double permutations appear twice in 36 rolls, for instance, 1:6 and 6:1.
• The chance of rolling a specific number on either die is 11 in 36, as there are 11 possible permutations that produce the number 1.
• Even though there are 36 permutations of the dice, only 21 combinations exist. For example, 6:5 and 5:6 are different permutations of the dice that result in the same destination for the moved piece.
• Do not fall into the trap of believing that the likelihood of rolling a specific number (like 6) in the next few turns increases because you have not rolled it in the last 10 turns. The dice have no memory and are inanimate. The odds of rolling a 6 always stay the same, at 11 in 36, whether you have just rolled sixes ten times in a row or haven’t rolled a six in the last ten turns.

The following table applies the dice combinations to the probabilities of hitting a blot. When studying this table, take note of the following:

• The riskiest place to leave a blot is six points away from your opponent. The chances of them hitting it are 17 in 36.
• If you have to expose a blot, it’s better to do so indirectly (i.e. with a number that requires the combined total of both dice to hit it). The farther away the blot is from your opponent, the better your chances of survival.
• If your blot is exposed to a direct shot (i.e. a number that is less than seven), try to place it as close to your opponent as possible. The closer it is to them, the lower the chances of it being hit.

### The Importance of Blots in the Opening Move of Backgammon

The diagram displays a number on top of each point, indicating the number of dice combinations your opponent can use to hit a white blot exposed on that point. It is evident that the three safest points to leave an opening move blot are 11, 10, and 9.

## Strategic Sacrifice in Backgammon

On the Backgammon board, point 7 is considered to be one of the riskiest spots to leave a blot as 24 different dice combinations can hit it. Even though it may seem unwise, many skilled players use this as their opening move. The reason behind this move is not just to expose the blot, but also to gain an advantage in the game.

If you leave a blot on point 7, there are several possible outcomes:

• 19 out of 36 times, your opponent will hit your blot and leave a blot themselves.
• 12 out of 36 times, your opponent will miss your blot.
• 5 out of 36 times, your opponent will hit your blot and cover their own blot.

This sacrifice move is particularly advantageous when you have three men separated in your opponent’s home board. However, you should avoid this move if your opponent has made points in their inner board, or if re-entry chances are reduced. The same argument applies to point 5, which is another vital point for your opponent.

The probability of re-entry is crucial to consider in Backgammon. For instance, if you have one man on the bar and your opponent has 3 closed points and 3 open points, your chances of re-entry are better than 50-50. Only 9 dice combinations can stop re-entry, making the odds 3 to 1 or 75%. However, if you have two men on the bar, your chances of re-entry fall dramatically to 25%. It is essential to avoid giving your opponent the opportunity to get two of your men on the bar, and to strive to get two of their men on the bar instead, even if it means paying a high price.

In the given diagram, White rolls a 6:1 and removes a man from point 5 with the 6. However, he is left with a 1 to play. He can either move a man from point 4 to point 3 or move a man from point 3 to point 2. Which move is better? It is observed that when two men are placed on point 3, there is a 47% chance of both being hit on the next roll. On the other hand, when the men are split and placed on points 4 and 2, the chances of at least one of them being hit rise to 64%.

In the same diagram, White rolls a 2:1 and has three options to choose from. He can either remove the man on point 1 and move one man from point 5 to point 3, or remove a man from point 2 and move a man from point 5 to point 4, or remove two men from points 1 and 2, leaving one man on point 5. Which option is the best? It is found that if White chooses the first option (3-2), he has a 69% chance of bearing off in one roll. If he goes with the second option (4-1), his chances increase to 81%, and if he selects the third option, his chances are the highest at 86%.

While it is not necessary to memorize all the probabilities, it is crucial to remember three key facts about the end play. Firstly, it is better to leave one man on a higher point than to split two men on lower points. Secondly, when you have two men left to bear off and have a choice of which man to move, it is better to move the lower one down rather than the higher one. Finally, it is advisable to avoid leaving two men on the same point and instead keep them separated if possible.

## Double arithmetic

It is commonly believed that if an opponent doubles you, then you must be in a position where you must refuse the double. However, this is not always the case. There are situations where it is appropriate to double and accept the double.

One such situation is depicted in the diagram. It is Black’s turn to roll the dice. He knows that he has a 64% chance of clearing both his checkers on the next roll, so he doubles White. White correctly accepts the double. Why? To make it clear, let’s assume that White found himself in this position 100 times in a row, and he was playing for 1€ per point. If he refused the double every time, he would lose over 100 x 1€ = 100€. However, if he accepted the double every time, the result would be:

• Lose 64 times (doubled) 64 x 2€ = 128€
• Win 36 times (doubled) 36 x 2€ = 72€
• Result (loss) = 56€

Even though he still loses over the 100 games, by accepting the double, he reduces his loss from 100€ to only 56€. The break-even point for accepting the double is 75% to Black, 25% to White. If the odds are worse than this, White is better off refusing. If the odds are better than 75/25, White will lose less by accepting.