Backgammon Odds

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Like chess, it is virtually impossible to come up with a formula for successful Backgammon play. There are simply too many variables involved, and the human element does not easily factor into any equation. However, because it is a board game played with dice, a finite number of playing pieces and only 24 points to land on, there are some calculations that can be useful in developing winning strategies.

Thankfully, the probabilities associated with rolling the dice at Backgammon are rather easy to understand. They are exactly the same as shooting at the Craps table. In total, there are just 36 possible combinations of “pips” (the dots on the dice)—less than the number of possible outcomes on a Roulette wheel.

The probability of any specific combination being rolled is exactly 1-in-36, or 2.78%. The probability of any specific total appearing is the sum of all the ways the total can be formed divided by 36. For example, if a player needs to move a blot seven points to safety or to fill a gap in a wall, there are six possible combinations (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), so the chances of seven being rolled are 6-in-36, equivalent to 1-in-6 or 16.67%.

Similarly, with a piece on the Bar and only the 1-point out of six available to reenter on, the likelihood of rolling the needed number one is not 1-in-6 or 1-in-3. It is the sum of the ways the number one can be rolled (1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 6-1, 5-1, 4-1, 3-1, 2-1) divided by 36, i.e., 11-in-36 or 30.6%. The same is true of any other point needed.

It must also be kept in mind that the number of points moved can be a direct number or an indirect number. For example, to bump off a blot three points ahead, a piece can reach the point directly by having a three appear on one of the dice (11 ways) or indirectly by rolling a total of three (1-2, 2-1) or double one (1-1). That’s 14 combinations out of 36 or 38.9%.

Moving a runner to safety, covering a gap, hitting a blot and reentering from the Bar are just a few of the situations in which calculating Backgammon odds can be applied. The probabilities can also be used for bearing off, determining how likely a safe roll is and, although more complex, optimum positioning of runners for defense or offense.

Experienced players memorize the chances of moving “n” points on a single roll. The easiest to remember are the first six because they involve direct moves only. The probability is exactly 31% for each number 1~6. With indirect moves included, the odds are roughly 31% for one point, 33% for two, 39% for three, 42% for four or five, and 47% for six.

For seven points or higher, all moves must be indirect, starting at 17% for seven, eight or nine, 8% for ten or twelve, and 6% for eleven. Above that, doubles are required, such as for point totals of fifteen, sixteen or eighteen, each of which has a likelihood of 3%. Some point totals, of course, are impossible to roll, so the odds are zero. That includes thirteen, fourteen, seventeen and any total above eighteen except twenty and twenty-four, which can be rolled with 5-5 or 6-6, respectively, at odds of 3%.

Why is it important to know such probabilities? When wagering with the doubling cube, the player must know the odds of the worst or best roll occurring. It is the only way to determine whether a double is worth making or accepting.

Published on: 08/06/2012

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