What Are The Chances?
â€œAnd if you think you could
Well, chances are your chances are awfully goodâ€
~ Johnny Mathis
Lyrics by Robert Allen and Music by Al Stillman
Backgammon is a game of risk. Playing it safe is often a sure path to long term disaster. Whether played for FIBS rating points, between friends some evening, in a tournament, or for money, backgammon is almost always an exciting unpredictable game where risk and reward go hand in hand. But there is a delicate balance between minimizing risk and maximizing reward.
Underlying the unpredictable risk and reward are the dice. Anything can and does happen whether we shake those bones and they dance out onto the table or the FIBS dice dramatically appear on the screen.
Weâ€™re all familiar with basic dice probabilities  thirtysix possible outcomes when rolling two six sided dice. When the two are combined as a total there are exactly 6 ways to roll a 7; 2 ways to roll an 11; and only 1 way to roll double sixes. One way in 36, that is.
Another way of saying this is â€œ1 chance in 36â€, or a â€œ1 in 36 chanceâ€ to roll double sixes. Six ways becomes a â€œ6 in 36 chanceâ€ or more simply â€œ1 chance in 6â€ or a â€œ1 in 6 chance.â€ As we know from watching thousands of rolls, one in thirtysix is fairly rare and almost always welcomed with relief or chagrin by one of the players. A one in six chance on the other hand is much more predictable and not so surprising.
We can and do expect a â€œ1 in 6â€ risk every time we hit an opponents blot on our 7point. When coming off the bar, our opponent has a 1 in 6 chance of rolling a seven and hitting us back and sending us back to the bar. If our 6point is the only point covered in our home board, there is a 1 in 36 chance reentry will be blocked.
Expressing the chances for moving or of being hit (assuming no intervening blocks points) for basic backgammon rolls is often written about and frequently expressed in percentages, three digit probabilities, or as a fraction, often in 36^{ths} such as 6/36. While interesting and precise, these numbers can seldom be recalled in the heat of battle during a game. But turn that 6/36 into 1/6 and there you have the essence of the chance expression â€" â€œ1 chance in 6â€ of being hit and conversely â€œ5 of 6 rollsâ€ will miss. Therein lay the beginnings of using dice probability in risk decision making.
While playing around with Excel recently, I discovered a more intuitive approach where the probabilities or odds of moving n points can simply be expressed as 1 chance in 2, or 1 in 3, or 1 in 6, and so on. When facing a difficult choice on where to move or your chances of being hit this is a readily remembered and more meaningful way of thinking. For readability, the odds or chances are expressed below as a fraction (n/m or n chances in m tries). Here is the summary:
Approximate Chances of Moving n Points in a Single Roll
Range 
1 
2 
3 
4 
5 
6 
7 
8 
9 
Chances 
1/3

1/3 
2/5 
2/5 
2/5 
1/2 
1/6 
1/6 
1/6 
10 
11 
12 
15 
16 
18 
20 
24 
1/12

1/18 
1/12 
1/36 
1/36 
1/36 
1/36 
1/36 
Some readers may immediately recognize this is not accurate. It is an approximation designed for ease of recall. It is, however, accurate within an acceptable margin of error, +/ 3%. This ease of recall makes it well worth the little effort required to burn into our brain or perhaps even print and tape to the monitor while you play. The remainder of this article supports this proposition.
For an excellent detailed overview of backgammon dice rolls, complete with an interactive display of the combinations required for any outcome, see the article in Paul&undefined;s Pages cited below. This article will not attempt to recreate that. But, Paul uses percents, one of the more traditional ways of expressing the probabilities of each roll. Compare the tables above and below for a moment. Which one are you more likely to remember? Which makes more sense for your decision making purposes in the heat of a game? Thereâ€™s at least a 1 out of 2 chance you may agree with me.
Chances of moving n points in a single roll.
Move 
Direct 
Direct & 

Move 
Indirect 

Move 
Indirect 

Move 
Indirect 
7 
17% 
13 
 
19 
 

1 
31% 
31% 
8 
17% 
14 
 
20 
3% 

2 
33% 
9 
14% 
15 
3% 
21 
 

3 
39% 
10 
8% 
16 
3% 
22 
 

4 
42% 
11 
6% 
17 
 
23 
 

5 
42% 
12 
8% 
18 
3% 
24 
3% 

6 
47% 
Source: http://www.paulspages.co.uk/bgvaults/tips/dicerolls.php#stats
While Paulâ€™s table is an accurate and comprehensive way of expressing dice probabilities, it is not something easily recalled other than intuitively while playing and trying to decide where to move or the chances of being hit on the next roll when offered a cube. Most of us make those decisions by the seat of our pants and far more goes into such decisions than these simple probabilities. Whether it helps my play or just provides a meaningful commentary on my risk taking decisions, I&undefined;m liking the feel of knowing my chances. So, I propose the use of my chance table gives us an easier way to knowledgably approach a game situation in the heat of combat.
As stated, the chance table is an approximation with a margin of error. The discussion below shows how it was derived. For readability below the odds or chances are expressed as a fraction (n/m or n chances in m tries). Where the chances have a +/ suffix the exact value is marginally more or less. I contend that worrying with the +/ suffix becomes unnecessary as it effectively means +/ 3% at most. In other words, the margin of error in these chance expressions is less than the chance of rolling double sixes, 1 in 36 or 3%. That is close enough for me.
Building a Chance Table Using Approximations

1 in 3 to 1 in 2 
1 in 6 

Range 
1 
2 
3 
4 
5 
6 
7 
8 
9 
Chances 
1/3

1/3 
2/5 
2/5+ 
2/5+ 
1/2 
1/6 
1/6 
1/6 
Percent

31% 
33% 
39% 
42% 
42% 
47% 
17% 
17% 
14% 
Exact 
11/36 
1/3 
7/18 
5/12 
5/12 
17/36 
1/6 
1/6 
5/36 
Ways in 36 possible outcomes 
11 
12 
14 
15 
15 
17 
6 
6 
5 
1 in 12 to 1 in 18 
1 in 36 

10 
11 
12 
15 
16 
18 
20 
24 
1/12 
1/18 
1/12 
1/36 
1/36 
1/36 
1/36 
1/36

8%

6% 
8% 
3% 
3% 
3% 
3% 
3% 
1/12 
1/18 
1/12 
1/36 
1/36 
1/36 
1/36 
1/36 
3 
2 
3 
1 
1 
1 
1 
1 
Or as a chance fraction in tabular form similar to Pauls example above:
Chances of moving n points in a single roll.
Move 
Direct 
Direct & 

Move 
Indirect 

Move 
Indirect 

Move 
Indirect 
7 
1/6 
13 
 
19 
 

1 
1/3

1/3 
8 
1/6 
14 
 
20 
1/36 

2 
1/3 
9 
1/6 
15 
1/36 
21 
 

3 
2/5 
10 
1/12 
16 
1/36 
22 
 

4 
2/5 
11 
1/18 
17 
 
23 
 

5 
2/5+ 
12 
1/12 
18 
1/36 
24 
1/36 

6 
Â½ 
In my opinion, trying to increase accuracy using percents is unnecessary and not too functional when making game decisions and can not be remembered, simplified fractions add accuracy but are also difficult to remember, even the +/ on my simplified chances is superfluous. Some may prefer to recall the purest form of probability in 36^{ths} and even be able to do something with them in obtuse calculations while playing. I canâ€™t other than intuitively.
Thus, the chances table , understood to be an approximation, in its final form becomes very easy to read:
Moving n Points in a Single Roll
Range 
1 
2 
3 
4 
5 
6 
7 
8 
9 
Chances 
1/3

1/3 
2/5 
2/5 
2/5 
1/2 
1/6 
1/6 
1/6 
10 
11 
12 
15 
16 
18 
20 
24 
1/12

1/18 
1/12 
1/36 
1/36 
1/36 
1/36 
1/36 
This I can recall. And Iâ€™ve found it to be a useful tool in assessing chances when taking risks in backgammon. Anyone with any experience in playing has understood this table intuitively, but Iâ€™m finding an increased level of confidence when understanding the chances of hitting that spot 6 points away is 1 in 2 (approximately). Hey, thatâ€™s a coin toss! Take a chance â€œAnd if you think you could, well, chances are your chances are awfully good.â€
As an afterthought, my color codes in the chance table may seem counterintuitive, as red traditionally means stop and here represents the greatest chances. I chose the redyellowgreen as a stoplight metaphor in an attempt to provide a defensive corollary within one table, i.e. red being the highest probability of being hit (1/3 to 1/2); yellow use caution (1/6); green more safety (1/12 to 1/18); and gray that gray area of uncertainty always present of being hit by boxcars or snakeeyes (1/36).
So, the next time you play and think: "What are the chances I will be hit?" visualize this table. Remember 1 out of 2, if 6 away; 2 out of 5, if 4 or 5 away; 1 out of 3, if 1 or 2 away; and so on. In no time it becomes second nature and may help in making placement decisions as well as in cubing.
Good luck and good dice ... when you need a 6, heres to rolling a sixty_something ... 8^)>
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