[GZG] The pain of dice stats

["Thomas Barclay" <kaladorn@xxxxxxxx>]

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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