Re: [DL] Dice Rollin'

["Johnny Burlin" <Johnny.Burlin@home.se>]

----- Original Message ----- 
From: Michael Sprague <mds@rochester.rr.com>
To: <deadlands@gamerz.net>
Sent: Saturday, March 15, 2003 2:20 PM
Subject: [DL] Dice Rollin'

<snip>

> Target#   1d10   2d10   3d10   4d10   5d10
> ------- ------- ------- ------- ------- -------
>  Bust 10.000%  1.000%  1.000%  0.100%  0.100%

<snip>

I hate to break this to you, I really do, but the chance of Bustin'
above is incorrect. It is correct for 1D10 and 2D10 but not
for 3D10 and above.

Think about it, who do you bust on 3D10? Well, whenever you
roll 1-1-X, where X can be any number. Huw bust rolls
can there be on a 3D10 roll? For starters X can be any of the
three dice and X is any number in the range 2-9, 1-1-1 is
a special case that is one of a kind, it shouldn't be counted
in all three combinations. So, you bust on the following
combinations:
1-1-1            1 combination
(2-9)-1-1      9 combinations
1-(2-9)-1      9 combinations
1-1-(2-9)      9 combinations
A grand total of 30 dice combinations results in busts.
There are 10^3  dice combinations so the chance of bustin' is 
30/1000 = 3%
Note that this is higher than the bust chance on 2D10, counter 
intuitive but a well known fact.

The bust chance on 5D10 is a little more complex, as before there
is exactly one "all ones" result. There is also 5 bust results with 1
"none one" result. Hmm, that last sentence got a little interesting
but it is correct, it means that you have 4 dice turning out as 1:s and
1 that is 2-9.
There are also 10 bust results with 2 "none one" results.
How I arrived at 5 and 10? Basic combinatorics, 
n over k = (n!/k!)/(n-k)!
5 over 4 = 5, 
5 over 3 = 10 

The following combinations are busts:
1-1-1-1-1                            1 combination

5 x (4 1:s, 1 (2-9))               5*9 = 45 combinations

10 x (3 1:s, 2 (2-9))             10*9*9 = 810 combinations
A grand total of 856 dice combinations results in busts
There are 10^5 dice combinations so the chance of bustin' is
856/100000 = 0.856%

While we are on the subject, did anyone notice that the chance of getting
3 action cards (Qui vs. 10) is higher when your Quickness is 4D8 than 
when it is 4D10 (assuming you suffer from no penalties), talk about 
counter intuitive...

/Johnny