Double attack and weapon skills

Previously, I estimated the average damage of both Raging Rush and King's Justice for my character on lv 82 greater colibri (link), but there was one major unmentioned assumption I made concerning how the double attack trait processes on weapon skills.

Suppose that "conventional wisdom" assumes that double attack can proc twice, at most, on a WS (I haven't seen any evidence to prove that DA can proc more than twice), but under this assumption there are two possibilities: (1) double attack must proc on only two hits of the WS (2 or more normal hits in the WS; this is usually thought of as occurring on the first two hits of the WS), and; (2) double attack may proc a maximum of two times on a WS. Which one is it?

There is a subtle difference between the two "hypotheses." If DA can proc on any hit in a multi-hit weapon skill, there are more opportunities for DA to proc twice (when the number of normal hits in the WS is greater than 2) than there would be if DA is limited to proc on specific hits in the WS. Intuitively, if the number of normal hits in the WS is greater than 2, there will be, on average, more WS hits under the second hypothesis even in the presence of a cap to exclude 3+ DA procs.

If you aren't convinced, the following probability exercise will help. Suppose I'm looking at a 3-hit WS (examples: Raging Rush, King's Justice, Blade: Jin, Tachi: Rana) and I want to know the probability of seeing n hits (n = 1, 2, ..., 5) in one WS, given my DA level. Assume 95% hit rate.

Since DA procs are independent of normal hits (in the sense that normal hits must occur in a WS even if they miss), it's simple to calculate these probabilities when DA must proc on only two hits in the WS. Here, the second DA proc is assumed to be independent of the first DA proc, and vice versa. For the other case, the DA procs are dependent, so the calculations are less simple, but they can be done.

When the DA rate is 10%, the probability distributions for both cases are illustrated as follows:



People are more likely to notice 5-hit results than other results, but in either case the probability of observing a 5-hit is pretty low. However, under "2 DA maximum" there are more opportunities for DA to proc (even if there is a 2-DA cap). The expected number of hits is 3.04 for "DA two hits only," and 3.13 for "2 DA maximum."

If you increase your DA rate, the expected number of hits for a WS should always increase (you will see relatively more 4- and 5-hit WSes), and this is the case going from 10% DA to 19% DA:



The expected number of hits is 3.211 for "DA two hits only," and 3.39 for "2 DA maximum." Given 19% DA, it is now fairly easy to distinguish between the two hypotheses, and collecting enough sample data on n-hits of a 3-hit WS should provide evidence in favor of one or the other.

If you can manage to push your DA rate even higher (through merits or elsewhere; I myself have 2 DA merits), the difference between the two hypotheses becomes more stark. Consider when DA is 22%:



The expected number of hits is 3.268 for "DA two hits only," and 3.47 for "2 DA maximum."

Which one do I believe to be the case? I don't have any stake in believing one over the other, but it was easier for me to assume that DA procs on two hits only (there are three ways this can happen for a 3-hit WS, but it doesn't matter in calculating the probabilities).