Originally Posted by Vulajin
I don't believe this is necessary because the 1/vp term already incorporates the average likelihood that swings do or not do not occur in the second immediately following the cooldown's expiry.
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Actually it does not. 1/vp is the correct average time until the next proc starting at time 0.
With uneven numbers, there is a gap until the next eligible attack. In essence, the cooldown is not actually 20 seconds but 20 seconds plus the time interval until you have another attack. And actually, looking at my notes, I see an error. It should be -1/2v not +1/2v. As the formula 1/vp is assuming a full interval until the next attack and what you are left with at the end of a cooldown is a partial interval.
In my example I'm representing that time interval as 1/2v which is the average time until another attack. In truth, if all attacks were perfectly clockwork, this factor would be something different (essentially whatever fraction of an attack that does not evenly fit into the cooldown period).
An easy way to test this is a simple example. You flip a coin 1 time per second. What is the average time until you get a heads? This can easily be calculated (and even charted) and the answer is 2 seconds. Now what happens if there is a 1.5 second cooldown until it "counts" a heads? The answer is 3 seconds not 3.5 seconds.