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Subject: Dice probability help for GG rss

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♫ Eric Herman ♫
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10 GG if someone can give me the percentages for the following list... I'm looking for the probabilities of getting combinations of ranges (i.e., 1-3, 4-5) on multiple dice, for a skill check type of thing.

EDIT: I found a few of these... but it's the combination of different ranges that I'd like some help with in particular.

2 dice: (1-3) + (1-3) = 25%
2 dice: (1-3) + (4-5)
2 dice: (1-3) + 6
2 dice: (4-5) + (4-5) = 11%
2 dice: (4-5) + 6
2 dice: 6 + 6 = 3%

3 dice: (1-3) + (1-3) = 50%
3 dice: (1-3) + (4-5)
3 dice: (1-3) + 6
3 dice: (4-5) + (4-5) = 26%
3 dice: (4-5) + 6
3 dice: 6 + 6 = 7%

4 dice: (1-3) + (1-3) = 69%
4 dice: (1-3) + (4-5)
4 dice: (1-3) + 6
4 dice: (4-5) + (4-5) = 41%
4 dice: (4-5) + 6
4 dice: 6 + 6 = 13%

5 dice: (1-3) + (1-3) = 81%
5 dice: (1-3) + (4-5)
5 dice: (1-3) + 6
5 dice: (4-5) + (4-5) = 54%
5 dice: (4-5) + 6
5 dice: 6 + 6 = 20%

6 dice: (1-3) + (1-3) = 89%
6 dice: (1-3) + (4-5)
6 dice: (1-3) + 6
6 dice: (4-5) + (4-5) = 65%
6 dice: (4-5) + 6
6 dice: 6 + 6 = 26%

7 dice: (1-3) + (1-3) = 94%
7 dice: (1-3) + (4-5)
7 dice: (1-3) + 6
7 dice: (4-5) + (4-5) = 74%
7 dice: (4-5) + 6
7 dice: 6 + 6 = 33%
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Paul Dale
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How about these:

2 dice: (1-3) + (1-3) = 1 / 4 = 25.0000 %
2 dice: (1-3) + (4-5) = 1 / 3 = 33.3333 %
2 dice: (1-3) + 6 = 1 / 6 = 16.6667 %
2 dice: (4-5) + (4-5) = 1 / 9 = 11.1111 %
2 dice: (4-5) + 6 = 1 / 9 = 11.1111 %
2 dice: 6 + 6 = 1 / 36 = 2.7778 %

3 dice: (1-3) + (1-3) = 1 / 2 = 50.0000 %
3 dice: (1-3) + (4-5) = 7 / 12 = 58.3333 %
3 dice: (1-3) + 6 = 1 / 3 = 33.3333 %
3 dice: (4-5) + (4-5) = 7 / 27 = 25.9259 %
3 dice: (4-5) + 6 = 1 / 4 = 25.0000 %
3 dice: 6 + 6 = 2 / 27 = 7.4074 %

4 dice: (1-3) + (1-3) = 11 / 16 = 68.7500 %
4 dice: (1-3) + (4-5) = 20 / 27 = 74.0741 %
4 dice: (1-3) + 6 = 101 / 216 = 46.7593 %
4 dice: (4-5) + (4-5) = 11 / 27 = 40.7407 %
4 dice: (4-5) + 6 = 31 / 81 = 38.2716 %
4 dice: 6 + 6 = 19 / 144 = 13.1944 %

5 dice: (1-3) + (1-3) = 13 / 16 = 81.2500 %
5 dice: (1-3) + (4-5) = 1085 / 1296 = 83.7191 %
5 dice: (1-3) + 6 = 185 / 324 = 57.0988 %
5 dice: (4-5) + (4-5) = 131 / 243 = 53.9095 %
5 dice: (4-5) + 6 = 215 / 432 = 49.7685 %
5 dice: 6 + 6 = 763 / 3888 = 19.6245 %

6 dice: (1-3) + (1-3) = 57 / 64 = 89.0625 %
6 dice: (1-3) + (4-5) = 581 / 648 = 89.6605 %
6 dice: (1-3) + 6 = 1687 / 2592 = 65.0849 %
6 dice: (4-5) + (4-5) = 473 / 729 = 64.8834 %
6 dice: (4-5) + 6 = 1729 / 2916 = 59.2936 %
6 dice: 6 + 6 = 12281 / 46656 = 26.3224 %

7 dice: (1-3) + (1-3) = 15 / 16 = 93.7500 %
7 dice: (1-3) + (4-5) = 43561 / 46656 = 93.3663 %
7 dice: (1-3) + 6 = 8323 / 11664 = 71.3563 %
7 dice: (4-5) + (4-5) = 179 / 243 = 73.6626 %
7 dice: (4-5) + 6 = 10423 / 15552 = 67.0203 %
7 dice: 6 + 6 = 7703 / 23328 = 33.0204 %

8 dice: (1-3) + (1-3) = 247 / 256 = 96.4844 %
8 dice: (1-3) + (4-5) = 16745 / 17496 = 95.7076 %
8 dice: (1-3) + 6 = 213781 / 279936 = 76.3678 %
8 dice: (4-5) + (4-5) = 5281 / 6561 = 80.4908 %
8 dice: (4-5) + 6 = 19219 / 26244 = 73.2320 %
8 dice: 6 + 6 = 663991 / 1679616 = 39.5323 %

9 dice: (1-3) + (1-3) = 251 / 256 = 98.0469 %
9 dice: (1-3) + (4-5) = 181405 / 186624 = 97.2035 %
9 dice: (1-3) + 6 = 37525 / 46656 = 80.4291 %
9 dice: (4-5) + (4-5) = 16867 / 19683 = 85.6932 %
9 dice: (4-5) + 6 = 48655 / 62208 = 78.2134 %
9 dice: 6 + 6 = 2304473 / 5038848 = 45.7341 %

10 dice: (1-3) + (1-3) = 1013 / 1024 = 98.9258 %
10 dice: (1-3) + (4-5) = 2473273 / 2519424 = 98.1682 %
10 dice: (1-3) + 6 = 8440421 / 10077696 = 83.7535 %
10 dice: (4-5) + (4-5) = 17635 / 19683 = 89.5951 %
10 dice: (4-5) + 6 = 3106939 / 3779136 = 82.2129 %
10 dice: 6 + 6 = 10389767 / 20155392 = 51.5483 %

11 dice: (1-3) + (1-3) = 509 / 512 = 99.4141 %
11 dice: (1-3) + (4-5) = 59737601 / 60466176 = 98.7951 %
11 dice: (1-3) + 6 = 13074743 / 15116544 = 86.4929 %
11 dice: (4-5) + (4-5) = 163835 / 177147 = 92.4853 %
11 dice: (4-5) + 6 = 17219543 / 20155392 = 85.4339 %
11 dice: 6 + 6 = 12909191 / 22674816 = 56.9318 %

12 dice: (1-3) + (1-3) = 4083 / 4096 = 99.6826 %
12 dice: (1-3) + (4-5) = 14996345 / 15116544 = 99.2049 %
12 dice: (1-3) + 6 = 107339687 / 120932352 = 88.7601 %
12 dice: (4-5) + (4-5) = 502769 / 531441 = 94.6049 %
12 dice: (4-5) + 6 = 59887373 / 68024448 = 88.0380 %
12 dice: 6 + 6 = 1346704211 / 2176782336 = 61.8667 %


- Pauli

Edit: extended the results up to 12 dice.
Edit the second: reduce fractions to their lowest terms.
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Alec
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Just skimmed, but noticed this mistake immediately.

Quote:
2 dice: (4-5) + (4-5) = 4 / 36 = 11.1111 %
2 dice: (4-5) + 6 = 4 / 36 = 11.1111 %


Those can't have the same probability. The 2nd one should be 2/36.
 
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Richard Linnell
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Texjets281 wrote:
Just skimmed, but noticed this mistake immediately.

Quote:
2 dice: (4-5) + (4-5) = 4 / 36 = 11.1111 %
2 dice: (4-5) + 6 = 4 / 36 = 11.1111 %


Those can't have the same probability. The 2nd one should be 2/36.


I think it should be 1/18............always reduce your fractions!
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Texjets281 wrote:
Just skimmed, but noticed this mistake immediately.

Quote:
2 dice: (4-5) + (4-5) = 4 / 36 = 11.1111 %
2 dice: (4-5) + 6 = 4 / 36 = 11.1111 %


Those can't have the same probability. The 2nd one should be 2/36.


Die A shows 6 (1-in-6 chance) while Die B shows 4 or 5 (2-in-6) = 2-in-36

plus

Die B shows 6 (1-in-6 chance) while Die A shows 4 or 5 (2-in-6) = 2-in-36

equals

4-in-36 chance.

You can make a classic probability table as well if you like visual proofs:


Die A
1 2 3 4 5 6
1 o o o o o o
D 2 o o o o o o
i 3 o o o o o o
e 4 o o o o o X
5 o o o o o X
B 6 o o o X X o



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PenumbraPenguin
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Texjets281 wrote:
Just skimmed, but noticed this mistake immediately.

Quote:
2 dice: (4-5) + (4-5) = 4 / 36 = 11.1111 %
2 dice: (4-5) + 6 = 4 / 36 = 11.1111 %


Those can't have the same probability. The 2nd one should be 2/36.


Paul is assuming that the dice are indistinguishable. If this assumption is valid, then he is correct. Looking at the question, I would agree with this interpretation, because if the dice are distinct then you need more information to describe the problem for more than 2 dice.

To get 4-5 on both dice, you can have (4,4), (4,5), (5,4) or (5,5), for 4/36.

To get 4-5 on one die and 6 on the other, you can have (4,6), (5,6), (6,4) or (6,5), for 4/36.

If you want to get 4-5 on the red die and 6 on the blue die, then you have only two possibilities (4,6) and (5,6), so your odds are 2/36.
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Paul Dale
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Nope, they are the same.


The four possibilities for (4-5) and 6 are:

4 6
5 6
6 4
6 5

There is no overlap or duplication so the final chance is 4 in 36


For (4-5) and (4-5) the possibilities are:

4 4
4 5
5 4
5 5

Again four of them.


- Pauli
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solidhavok wrote:
I think it should be 1/18............always reduce your fractions!


I know that this is not the point, but I'm a college math instructor, and...

NO! DON'T REDUCE THOSE FRACTIONS!!

It is much clearer to write 2/36 than 1/18. 2/36 clearly says "out of all 36 ways to roll the dice, 2 of them give the result I want", whereas 1/18 says... nothing, at least until you UN-reduce it. Similar for cards: Might as well leave everything over 52, because that's the most useful value anyhow.
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Richard Linnell
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Actually, in several card games, it is best to have the reduced fraction. Particularly in Poker, where the odds of what you can get need to be balanced against the ratio of money that you are going to have invested vs. the amount of money in the pot. Even without talking about the relative payoff vs. the odds, knowing that there are 3 cards out of 51 just seems clunkier to me than saying 1 in 17.

It also is easier for many people to digest smaller fractions that are more familiar to them, and generally easiest (IMHO) to digest percentages.

1/18 says exactly what it means to say - one out of every eighteen rolls will yield the desired result, on average. Which, of course, is the exact same thing as saying 2/36.
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