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Date: Mon, 24 Oct 2005 17:24:55 -0400

Subject: [GZG] The pain of dice stats

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By exhaustive cases
Yielding 19:
D1: Nothing
D2: 19 points
D1: 1 points
D2: 18 points
D1: 2 points
D2: 17 points
D1: 3 points
D2: 16 points
etc.
and then back down the other side
D1: 9 points
D2: 10 points
D1: 10 points
D2: 9 points
D1: 11 points
D2: 8 points
etc. downto
D1: 19 points
D2: Nothing
That would cover all possible combinations leading to 19 points.
Then, it wouldn't be that hard to determine the possible ways to
generate each of these
scores per die. Usually, there is only one way.
The sum of these chances vs. the sum of all possible outcomes on two
dice will tell you
how likely things are to happen.
Using a little Excel wizardry
D1 Rslt D2 Rslt Odds D1 Odds D2 Combined Odds D1 + D2
0 19 0.5 3.27456E-08 1.63728E-08
1 18 0.33 9.9229E-08 3.27456E-08
2 17 0.166666667 1.96473E-07 3.27456E-08
3 16 0.055 5.95374E-07 3.27456E-08
4 15 0.027777778 1.17884E-06 3.27456E-08
5 14 0.009166667 3.57225E-06 3.27456E-08
6 13 0.00462963 7.07305E-06 3.27456E-08
7 12 0.001527778 2.14335E-05 3.27456E-08
8 11 0.000771605 4.24383E-05 3.27456E-08
9 10 0.00025463 0.000128601 3.27456E-08
10 9 0.000128601 0.00025463 3.27456E-08
11 8 4.24383E-05 0.000771605 3.27456E-08
12 7 2.14335E-05 0.001527778 3.27456E-08
13 6 7.07305E-06 0.00462963 3.27456E-08
14 5 3.57225E-06 0.009166667 3.27456E-08
15 4 1.17884E-06 0.027777778 3.27456E-08
16 3 5.95374E-07 0.055 3.27456E-08
17 2 1.96473E-07 0.166666667 3.27456E-08
18 1 9.9229E-08 0.33 3.27456E-08
19 0 3.27456E-08 0.5 1.63728E-08
Total Results leading to 19 points: 6.22166E-07
In all of these cases, once you break the two dice apart, their is
precisely one way to
generate the point score with one die roll sequence. To get 1, I need to
roll 4 or 5 (ergo
0.5). To get 2, I need to roll 6. To get 3, I need to roll 6 followed by
4 or 5, etc.
Treating each dice separately, and multiplying the probabilities of a
given outcome
together should give us the total for that particular combination of
points.
Then sum these up for the total. I make this at 1 in 1607288.038, given
some rounding
errors. So call it 1 in 1.6 million in round figures.
Pretty unlikely.
Can we put this bad boy to rest now?
Regardless of the math, it's a pretty freakin' unlikely circumstance and
to make rules
from it seems a poor choice. (YMMV).
TomB
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