## Backgammon

Okay, so there are a lot of backgammon pages out there. I'm just adding this because I wanted to show the probabilities to a friend, so I might as well post them for all to see.

It is critical to determine probabilities of certain moves taking place, however, because each die is a separate move, it is necessary to consider all possible cases.

Normally, most people are aware that the probabilty of getting a 7 on a roll of two die is 1/6. However, the probability of getting a 6 as some combination is significantly higher—17/36 or almost 50%. All combinations which contain at least one move of 6 include:

 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

Figure 1. The rolls resulting in a move of 1.

 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

Figure 2. The rolls resulting in a move of 2.

 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

Figure 3. The rolls resulting in a move of 3.

 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

Figure 4. The rolls resulting in a move of 4.

 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

Figure 5. The rolls resulting in a move of 5.

 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

Figure 6. The rolls resulting in a move of 6.

Some moves have no possible combinations:

13, 14, 17, 19, 21, 22, 23, 25, ...

What makes the game more interesting is when you have a number of covered points in front of the piece you are intereted in moving.

Anyway, the next three graphs show the probabilities of certain moves as a result of the given combinations, with the last figure giving to sum of the probabilities. Each square represents a probability of 1/36.

Here is a pdf of the four charts and you should also consider visiting Wikipedia's Backgammon page.