Backgammon, Part 4

Risk.
The chance that your opponent may land on one (or more) of your blots and send it (them) to the bar.

Backgammon is all about probabilities. It plays a huge role in the game. First, let’s look at the dice.

The game is played with two 6-sided die, that are numbered 1 though 6, each. Each die has the same chance of coming up as a 1, as it does coming up as a 6. The die are unrelated to each other, so getting a 1 on the first die has no impact on whether the second one also comes up as a 1. And, the previous rolls have no influence on the next roll, so if you get double 6’s before, you still have the exact same odds of getting double 6’s next time, too.

If we chart out the possible permutations, we get:
1-1, 1-2, 1-3, 1-4, 1-5, 1-6
2-1, 2-2, 2-3, 2-4, 2-5, 2-6
3-1, 3-2, 3-3, 3-4, 3-5, 3-6
4-1, 4-2, 4-3, 4-4, 4-5, 4-6
5-1, 5-2, 5-3, 5-4, 5-5, 5-6
6-1, 6-2, 6-3, 6-4, 6-5, 6-6

There are 36 possible outcomes, but a little under half of them are identical as far as the game is concerned. That is, a 1-4 is the same as a 4-1. The order doesn’t matter.

So, if we want a 1-4 (or 4-1), the chances are 2 out of 36, 1 out of 18, or 5.6%. Put another way, out of 100 rolls, only 5 of them will be specifically a 1 combined with a 4. That’s not significant enough to plan a game around.

If we want double 1’s, that’s 1 out of 36, or 2.8%

However, if all we want is any kind of doubles (double 1’s, double 2’s, whatever), that’s 6 out of 36, or a 1/6th chance, at 16%. So, for a running game, where the player that gets the most doubles will win, you can expect doubles every sixth role, which is almost guaranteed to happen at least once during the running game.

If we add the numbers up, we find that there’s only one way to get a 2 (1-1), two ways to get a 3 (1-2, 2-1), three ways to get a 4 (1-3, 3-1, 2-2), etc.

2 – 1 – 2.8%
3 – 2 – 5.6%
4 – 3 – 8.4%
5 – 4 – 11.2%
6 – 5 – 14%
7 – 6 – 18%
8 – 5 – 14%
9 – 4 – 11.2%
10 – 3 – 8.4%
11 – 2 – 5.6%
12 – 1 – 2.8%

So, while there’s only one way to get a total of 2, there are 6 ways to get a total of 7. Compare 2.8% versus 18%. That means that if your opponent’s blot is 2 points away from you, you only have a 2.8% chance of hitting it, right?

Well, not exactly. In backgammon, you move the number of points indicated on EACH die. So, when you’re figuring the chances of being hit, or hitting your opponent’s blot, you need to keep in mind the odds of getting all combinations of a specific number. So, if your opponent is 2 points away, you want a 2 on one die, or any combination that adds up to 2. (Any number followed by a 2, or a 2 followed by any other number, plus double 1’s).

That would be: 1-2, 2-2, 3-2, 4-2, 5-2, 6-2, 2-1, 2-3, 2-4, 2-5, 2-6, 1-1

12/36 = 33%.

Them’s good odds. Essentially, one out of every three rolls will give you a 2. The odds get better if you’re looking for a 6. (16/36 = 44%). Things get much weaker above 7, though. 8: 5/36; 9: 4/36; 10: 3/36; 11: 2/26; 12: 1/36.

This means that if your opponent is close to your blot (between 1 and 7 points away), you have a 30% to 40% chance of getting sent to the bar. If you’re looking at switching to a running game and you need to move one stone 12 points to get to your home field, you have a 2.5% chance of doing that in 1 roll (it’s more likely to take 2 to 3 turns to move one stone 12 points.)

(Really risky starting move for 3-4.)

If we look at the example from last week, it’s the beginning of the game, you’re moving first, and you rolled a 3-4. If you take one stone from 13 and move it to 10, and another from 13 to 9, you have 2 open blots – how likely is it that White can hit you?

Well, the only stones White has that endanger you are on 1. That means White needs an 8 or a 9. For an 8, that’s 6-2, 5-3, 4-4, 3-5 and 2-6. For a 9, that’s 6-3, 5-4, 4-5 and 3-6. 9/36 = 25%. So, you can probably get away with this gambit 3 out of 4 games where you start with a 3-4. Otherwise, when White DOES get 2-6 or 3-5, he’ll most likely land on the 9 point and send you to the bar. That’s not all that damaging, though, this early in the game.

So, this is one way of measuring risk, the statistical odds that a specific blot can be hit by the opponent. Let’s look at this more next time.

To be continued.

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