Yeah, that was my thread. The original indeed only got a reasonable, but not truly insane number of attacks most of the time... maybe. The later version absolutely went to infinity.
It was never actually proved that the original went to infinity... depending on your modeling method, it either averaged at infinity or never went infinite... we never did actually sort that one out. It involved a lot of higher math to even try to figure it out, but we never actually did. Some people talked about modeling it, but they never actually did so.
JaronK
For your original build, the probability of gaining one attack at any point is 0.5225. The probability of losing one attack is 0.45. For simplicity, let's assume the non-crit threats lose us an attack. Thus:
[spoiler=Wall of Math]
Pr(Terminates after X crits) =
9+XC
X(0.4775)
(9+X)(0.5225)^
X= 1.29 x 10
-3 x (9+X)!/(9!(X!)) x (0.4775)
X(0.5225)
X= 1.29 x 10
-3 x (10)(11)...(9+X)/X! x (0.4775 x 0.5225)
X= 1.29 x 10
-3 x (10)(11)...(9+X)/X! x (0.2495)
XAs X becomes large, the second term, with the factorials, increases very slowly (As X becomes large, (X+9)/X => 1).
More specifically, for X>3, the term increases by less than a factor of 4. Since the third term applies a factor less than 1/4, the function is monotonically decreasing for X>3. For X=4, the function drops by a factor less than (13/4)x(1/4)=13/16. So for X>= 4, Pr (T(X)) <= Pr (T(4)) x (13/16)
(X-4). We can do this with series directly; the probability of terminating with four or more crits is bounded by (16/3)Pr(T(4)).
Therefore, the series as a whole is bounded by Pr(T(0)) + Pr(T(1)) + Pr(T(2)) + Pr(T(3)) + (16/3)Pr(T(4)).
Pr(T(0)) = 0.4775
9 = 1.29x10
-3Pr(T(1)) = 1.29 x 10
-3 x 10 x 0.2495 = 3.22x10
-3Pr(T(2)) = 1.29 x 10
-3 x (10)(11)/2 x (0.2495)
2 = 4.42x10
-3Pr(T(3)) = 1.29 x 10
-3 x (10)(11)(12)/6 x (0.2495)
3 = 4.41x10
-3Pr(T(4)) = 1.29 x 10
-3 x (10)(11)(12)(13)/24 x (0.2495)
4 = 3.58x10
-3Therefore, our loose upper bound for finite attack results is 0.191 + 4.41x10
-3 + 4.42x10
-3 + 1.29x10
-3 = 0.0292, or almost 3%. Thus it doesn't terminate over 97% of the time.[/spoiler]
Is that a detailed enough analysis?