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:::::::: Cedges, could you please actually provide a proof that this series is divergent? You keep saying it, but that's a statement that actually requires a rigorous proof. It will certainly lay that argument to rest; alternatively, you could talk about the ''mean'' and ''median'' damage per round -- show that the calculation of such leads to an infinite number (countable or otherwise). Plus, you cannot threaten a creature with criticals who is immune to criticals. That is complete nonsense. [[User:Surgo|Surgo]] 19:14, June 19, 2010 (UTC)
=== Divergence of new build for Attack - AC of -2 vs critical immune foe ===
This is a random walk beginning at 1 and terminating when 0 is reached. The position represents the number of remaining attacks. It is decreased by one each time step, and increased by one for each granted attack.
At each step, there are the following possible outcomes, with change in number of pending attacks (including making this attack), and the probability that they happen
* -1 1/20 Natural one on Two with One Blow
* -1 4/20 No threat, miss foe
* Varies 9/20 No threat, hit foe, resulting in Chakram ricochet against ally
** -1 1/20*9/20 Natural one vs ally
** -1 13/20*9/20 Hit ally with no threat
** Varies 6/20*9/20 Threaten ally on ricochet
*** 0 1/20*6/20*9/20 fail to confirm vs. ally
*** 1 19/20*6/20*9/20 confirm crit vs ally
* Varies 6/20 threat, hit foe resulting in chakram ricochet against ally
** Varies 1/20*6/20 fail to confirm TwOB threat, continue with ricochet:
*** 0 1/20*1/20*6/20 Natural one vs ally
*** 0 13/20*1/20*6/20 ally with no threat
*** Varies 6/20*1/20*6/20 Threaten ally on ricochet
**** 1 1/20*6/20*1/20*6/20 fail to confirm vs. ally
**** 2 19/20*6/20*1/20*6/20 confirm crit vs ally
** Varies 19/20*6/20 confirm TwOB threat, continue with ricochet:
*** 1 1/20*19/20*6/20 Natural one vs ally
*** 1 13/20*19/20*6/20 ally with no threat
*** Varies 6/20*19/20*6/20 Threaten ally on ricochet
**** 2 1/20*6/20*19/20*6/20 fail to confirm vs. ally
**** 3 19/20*6/20*19/20*6/20 confirm crit vs ally
This results in the following total probabilities for the 5 outcomes: -1: 113/200, 0: 69/4000, 1: 13119/40000, 2: 171/20000, 3: 3249/40000. In 40000ths they are -1: 22600, 0: 690, 1: 13119, 2: 342, 3: 3249. The average drift each step is (-22600+0*690+13119+2*342+3*3249)/40000, or 950/40000, or 19/800. This is a random walk with bias or wind, the bias is away from zero in the same direction as the initial condition, and the step sizes don't increase. Therefore the average number of time steps until zero-crossing diverges. I need to find or write a proof of this obvious and simple property for walks with more than just a left or right step, failing that I'll work out the whole darned stochastic process. -[[User:Cedges|Cedges]] 03:34, June 21, 2010 (UTC)
