Backgammon Q: odds when 1 distant away from enemy

[mattw]

In The Backgammon Book by Jacoby and Crawford, Penguin Books, 1976, p. 88, the authors state that being hit from 1 distance away is 31%. These guys were world class players.

I have had a couple of stats/probability classes in school many years ago. I know enough to be dangerous to myself.

Here's the question: the probability of a one on one die is 1/6. The probability of a one on the second die is 1/6. Added together, the probability of a one on either dice is 1/6 + 1/6 = 2/6=1/3=33.3%. Why do the authors state the chance of being hit is 31%?

A difference of 2.3% may be inmaterial but I am curious. If any answers contain mathmatical calculations, please include a narrative.

Thanks in advance.

[LetYouDown]

36 combinations when you roll two dice. Combinations containing a 1 in bold:

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

11/36 or 30.555...%, rounded to 31.

You're running into a common problem. Look at it this way. Using your logic, if you roll 6 dice, you'll end up with a 100% chance of rolling a 1. This obviously isn't correct.

[mattw]

dang, thanks LYD

[AaronBrown]

In addition to LetYouDown's excellent response, I give you another way of looking at it.

When you add the 1/6 and 1/6, you're double counting the times you get one on both dice. So you need to subtract 1/36. 1/6 + 1/6 - 1/36 = 11/36 = 31%.

[mattw]

dang, thanks AB.

[BruceZ]

[ QUOTE ]
In addition to LetYouDown's excellent response, I give you another way of looking at it.

When you add the 1/6 and 1/6, you're double counting the times you get one on both dice. So you need to subtract 1/36. 1/6 + 1/6 - 1/36 = 11/36 = 31%.

[/ QUOTE ]

Or 1/6 + (5/6)*(1/6) = 30.56%. That is, the chance that you hit on the first die, plus the chance that you miss on the first die AND hit on the second die.

OR 1 - (5/6)*(5/6) = 30.56%. That is, 1 minus the probability of missing on both dice.