Dominions Dice Roll Chart

[alexti]

Originally posted by Saber Cherry:

quote:

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Originally posted by alexti:
Interesting... The chart doesn't exactly match the numbers I got myself. The numbers are quite close though. I would post my formula, but I can't even think of trying to type in HTML

Well, they're the same to 4 digits, which is about all I'd expect with 200 million rolls.

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Maybe the difference is down to the random number generator you're using. Pseudo-random number generator does not give exactly correct distribution, I wonder if you'd get a bit different results with another RNG.

You actually have a statistical formula that predicts the outcome, without simulation? That would be quite useful and speed up the combat simulator a lot

Yes, here's it:

A(m): { n1, n2: n1+n2=m, where n1,n2 independent illwinter dice rolls }
pA(m) = (q^(u+1))*{u(6-v(1-q)-q)+q(v-1)}, where u and v, such that m=5u+v and, u>=0 and v=1..5

B(m): { n1, n2: n1+n2>=m, where n1,n2 independent illwinter dice rolls }
pB(m) = (q^(u+1)*{u(20-5(v-1)(14-v)/12)+6-(v-1)(v-2)/12}

D(m): { n1, n2, n3, n4: n1+n2>=n3+n4+m, where n1,n2,n3,n4 independent illwinter dice rolls }
pD(m) = sum[s=1..5-v]f[s,u,s+v] + sum[s=6-v..5]f(s,u+1,s+v-5)
where u and v, such that m=5u+v and, u>=0 and v=1..5
and f(s,u,w) = (q^(u+2)){c2(q^2)*a1(s)*b2(u,w)+c1(q^2)*(a1(s)*b2( u,w)+a2(s)*b1(w))+c0(q^2)*a2(s)*b2(u,w)},
and
a1(s) = 6-s(1-q)-q
a2(s) = q(s-1)
b1(w) = 20 - 5(w-1)(14-w)/12
b2(u,w) = u*b1(w) + 6 - (w-1)(w-2)/12
c0(q) = 1/(1-q)
c1(q) = q/((1-q)^2)
c2(q) = (q+q^2)/((1-q)^3)

Hard to write formulas here I had to be creative in a few places. Hopefully it's understandable. For interested, I was just writing sequences of probabilities of various outcomes, writing roll results in 5u+v form and then getting summing by v in from of summing by u, summing by u is from 0 to infinity so I could roll it in (see c0,c1,c2 constants) and only leave summing from 1 to 5.

If you want I can send you code that implements it too.

Oh... also... it seems that the rounding of the Last digit may differ... all my numbers are rounded up for 5-9, and down for 0-4.

I'm using double float precision (15 digits) and only round when outputing the text, rounding is mathematical (same as yours).

P.S. Edited: fixed missed parametrization of Ci(q)

[ January 01, 2004, 05:58: Message edited by: alexti ]