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Backgammon Probabilities

/assets/images/inner/backgammon.gifEqual parts luck and skill, the game of backgammon is attractive to all kinds of people. In other words, it's possible for a newcomer to even beat a pro if he/she gets lucky with the dice. Such precariousness can't be found in other games such as chess or checkers. Of course, an experienced player will nearly always emerge as the victor in any series of games, but regardless, the luck factor is what draws many new players to backgammon.

Accordingly, in order to improve as a player, one must understand this luck factor. By comprehending the numbers involved and knowing the correct chances of a certain number coming up, you place yourself in a better position to make the correct decisions necessary for victory.

Your checkers are often exposed to the possibility of being hit. A player who understands the probability, and can account for the degree of risk, is in a better position to decide when and where to expose such pieces. "Odds" (a.k.a. 'probability') are defined as the ratio of probability that something will occur. Put differently, at some point in a game a player might discover that a pair of 5s is exactly what is needed to execute some plan, or capitalize on an opponent's vulnerability. The odds of rolling a pair of fives are calculated as follows:

There is a 1 in 6 chance of a rolled die of showing any one digit. So, in order for any two particular digits to be rolled, you must multiply the odds of each event occurring. For example, 1/6 X 1/6 = 1/36, or 35 to 1.

As you can see, the chances of rolling double-5's are not very high and, consequently, you shouldn't count on this occurring.

Possible Outcomes

A die has 6 sides and so, 6 possible outcomes, ranging from 1 to 6. When rolling two dice, you'll have 36 possible outcomes, (6 X 6 = 36) which are referred to as dice combinations.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

As you can see from the 36 dice combinations, the chances of rolling any particular doublet (for example double-fives) are the same; 1 in 36, or 35 to 1. (NOTE: The combinations in blue are the desired outcomes).

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

What if, however, you need two different numbers, for example a 5 and a 2? Have a look at the table below to determine your odds for rolling this:

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Now, there are two possible dice combinations with which you can achieve your goal. Your odds of hitting this combination have thus doubled; 2/36, or 17 to 1.

Along the same lines, consider the following: at an important stage of the game, you find yourself exactly 7 pips away from a crucial blot, with none of your opponent's checkers separating your men and the exposed blot. What are the odds of rolling the necessary digits now?

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Here you can see the odds have risen substantially in your favor, compared to the previous two scenarios. You now have a 6/36 (5 to 1) chance of rolling a seven.

Now, consider a situation in which you need only one particular digit to execute a plan, for example a one. Examine the table to determine the odds.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Judging from the table, we see that your chances of rolling a one (or any particular digit) are 11/36, or 30.6%

The final scenario to consider occurs when you are within 6 pips of an important position, and how you roll the dice plays no role in how you get to that position, whether it be a combination or a single digit. For example, one of your checkers is 5 pips from a crucial point (or blot). Look at the table to determine the odds of rolling the needed number.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

So, as you can see here, the odds of rolling the numbers necessary to move one of your checkers 5 pips are 15/36, or 41.7%

Odds of Re-entering From the Bar

Below, we've indicated the odds for rolling the correct dice combinations, which bring your checker back into the game, depending on the no. of points covered by your opponent. Remember this is when you have only one checker on the bar; when you have more than one the odds of re-entering two checkers increase significantly.

Number of Blocked Points Odds for possible re-entry
0 36/36
1 35/36
2 32/36
3 27/36
4 20/36
5 11/36
6 0/36

 

How to Use the Odds

You can employ the odds in any number of ways. Backgammon is a game that requires a lot of decisions. A solid understanding of the possibility of any particular roll occurring will give you a huge advantage over your opponent. To illustrate this, consider the following scenario:

It's the late stages of a match and you have three checkers remaining: one each on the 2-, 4-, and 6-points. Your opponent has three checkers on his 1-point. You roll a 6 and a 1. Now, you would bear off the checker on the 6-point, but what do you do with the extra 1? Checkers on 3 and 2, or checkers on 4 and 1? A quick examination of the odds is all that's necessary to make the best decision. If you leave checkers on 4 and 1, then you know that all you need to do is roll a 3, 4, or a 5 on my next roll, or doubles 2 or 3. The table illustrates this below:

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

As you can see, there are 29/36 combinations (80.1% chance) that will result in victory. Not bad. Now, let's consider the alternative; leaving checkers on the 3- and 2-points.

1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6

Here the chances of getting a good roll are only 25/36 or 69%, so it is clear that the first option is better. If you think about it in percentages, by leaving the remaining checkers on the 4- and 1-point rather than the 3- and 2-point, you have increased your chances of winning by 11%!

As you can see, being familiar with the odds, which takes some practice, can make decision-making much easier and fruitful!

General Guidelines

You can't draw or use a table like the ones above every time you need to determine the odds of any one combination occurring. So, a better idea is to devise and remember a shortcut that will allow you to make quick and accurate decisions. Check out the following breakdown:

The odds of rolling any particular number are:

1 (i.e., a 3) 11/36 (31%)
2 (i.e., a 5 or a 6) 20/36 (56%)
3 (i.e., a 4, 5, or 6) 27/36 (75%)
4 32/36 (89%)
5 35/36 (97%)
6 36/36 (100%)

Does that make it a bit simpler???

Calculating the Probabilities Hitting the Right Combination

This quick formula can be used in any situation where you are concerned with the probability of rolling productive numbers. This type of knowledge will help in your doubling decisions.

Chances of a successful roll = (the total of useable combinations/36) %

Whenever the useable combinations are greater than 18/36, then the probability is more than 50% that the outcome will be good for you. So, in this case, the player will know that there is a more than even chance of an immediately successful roll and that over a period of time (a game, for example) successful outcomes must occur more often than not.

With all of this in mind, you can see that your chances of success in backgammon can be greatly improved with a solid understanding of the odds involved. And, hopefully, you've seen here that such calculations aren't too difficult to make and with a bit of practice, such an understanding can easily be added to your bag of skills.

 

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