Kraken Club: DW Hit Distribution (by rounds)

[RyaWHM]

I wanted to get this started so the wonderful math experts here can help kind of validate this before I spend any more time on it. I just did some stuff in Open Office in like 15min but it's pretty neat to know. In particular, I wanted to figure out just how little STP helps in small doses (obviously larger amounts would have a better reliability of increasing speed to 100TP).

Main: Single-hit Weapon
Sub: Kraken Club
# of swings - grouped by # of rounds

6 rounds (117649 combinations)

 12 thru 19
 0.35% < 20
 0.53% = 20
 1.09% = 21
 1.98% = 22
 3.26% = 23
 4.87% = 24
 6.67% = 25
 8.40% = 26
 9.79% = 27
10.58% = 28
10.64% = 29
 9.99% = 30
 8.77% = 31
 7.20% = 32
 5.53% = 33
 3.98% = 34
 2.68% = 35
 1.69% = 36
 0.99% = 37
 0.54% = 38
 0.49% > 38
 38 thru 48


5 rounds (16807 combinations)

 10 thru 15
 0.32% < 16
 0.58% = 16
 1.29% = 17
 2.46% = 18
 4.13% = 19
 6.19% = 20
 8.35% = 21
10.22% = 22
11.40% = 23
11.68% = 24
11.01% = 25
 9.58% = 26
 7.70% = 27
 5.73% = 28
 3.94% = 29
 2.50% = 30
 1.47% = 31
 0.79% = 32
 0.39% = 33
 0.27% > 33
 34 thru 40


4 rounds (2401 combinations)

 8 thru 11
 0.25% < 12
 0.60% = 12
 1.50% = 13
 3.08% = 14
 5.35% = 15
 8.03% = 16
10.59% = 17
12.41% = 18
13.03% = 19
12.36% = 20
10.62% = 21
 8.29% = 22
 5.88% = 23
 3.79% = 24
 2.21% = 25
 1.16% = 26
 0.54% = 27
 0.33% > 27
 28 thru 32


3 rounds (343 combinations)

 6 or 7
 0.13% < 8
 0.53% = 8
 1.65% = 9
 3.86% = 10
 7.13% = 11
10.75% = 12
13.65% = 13
14.93% = 14
14.28% = 15
12.04% = 16
 8.96% = 17
 5.89% = 18
 3.41% = 19
 1.73% = 20
 0.74% = 21
 0.35% > 21
 22 thru 24


2 rounds (49 combinations)

 0.25% = 4
 1.50% = 5
 4.75% = 6
10.00% = 7
15.25% = 8
18.00% = 9
17.25% = 10
14.00% = 11
 9.75% = 12
 5.50% = 13
 2.50% = 14
 1.00% = 15
 0.25% = 16


1 round (7 combinations)

 5.00% = 2 
15.00% = 3
25.00% = 4
25.00% = 5
15.00% = 6
10.00% = 7
 5.00% = 8

Store TP: Helpfulness Formula

Can someone help me make a rough formula for evaluating a change in x-hit build? For example, take 15-hit vs 14-hit from adding Rajas or something. How can I figure out the likelihood of landing on each (exactly) to weigh against the ~7.14% increase in WS frequency?

[Motenten]

Check one of the melee pages on my DPS spreadsheets.

Essentially you progress in four stages:
1) Probability of a given number of swings per round (dependent on OAT/DA/TA/etc; yours will be slightly more complicated)
2) Probability of a given number of hits per round given a particular number of swings (dependent only on acc, since zanshin shouldn't be a factor here)
3) How many rounds will it take (on average) to land a specific number of hits given the distribution of (2) (the avg rounds column on the right)
4) (from my weaponskill pages) How many hits do you need to land to reach 100 TP after any given distribution of TP return on the weaponskill

[Raelia]

Would depend on mainhand TP/hit as well.

[RyaWHM]

Would depend on mainhand TP/hit as well.

The exact TP per hit shouldn't matter because I'm not evaluating individual points of STP. You can figure that out separately and decide if it makes gear sense. The goal here is to try to figure out the probabilities of two x-hits (so spanning many delays).

Disclaimer: I am also ignoring accuracy, which shouldn't change outcomes very much for what I'm trying to figure out. If there's a hugely important reason to account for ACC then please tell me! (thankums!)

For example:
Let's say [weapon]+KC is a 16-hit average from WS to WS. Odds calculations are taken from my previous post: x hits, sums of GE's vs LT's x, LT advances to next column (repeat). Now we walk this down the tiers of STP benefits (lowering x-hit build).

16-hit:
3 rounds: 33% chance
4 rounds: 60% chance
5 rounds: 7% chance
(average 3.74 rounds)

15-hit:
2 rounds: 1% chance
3 rounds: 47% chance
4 rounds: 49% chance
5 rounds: 3% chance
(average 3.54 rounds)

14-hit:
2 rounds: 4% chance
3 rounds: 60% chance
4 rounds: 35% chance
5 rounds: 1% chance
(average 3.33 rounds)

13-hit:
2 rounds: 9% chance
3 rounds: 69% chance
4 rounds: 22% chance
(average 3.13 rounds)

12-hit:
2 rounds: 19% chance
3 rounds: 70.5% chance
4 rounds: 10.5% chance
(average 2.91 rounds)

11-hit:
2 rounds: 33% chance
3 rounds: 63% chance
4 rounds: 4% chance
(average 2.71 rounds)

10-hit:
2 rounds: 50% chance
3 rounds: 49% chance
4 rounds: 1% chance
(average 2.51 rounds)

9-hit:
2 rounds: 68%
3 rounds: 32%
(average 2.32 rounds)

Now if you were to run these x-hits for KC's "dummy" average (3.82 swings; so 4.8 w/DW), you get:
16-hit: 3.33 rounds (vs 3.74 real, from above)
15-hit: 3.13 rounds (vs 3.54 real, from above)
14-hit: 2.92 rounds (vs 3.33 real, from above)
13-hit: 2.70 rounds (vs 3.13 real, from above)
12-hit: 2.50 rounds (vs 2.91 real, from above)
11-hit: 2.29 rounds (vs 2.71 real, from above)
10-hit: 2.08 rounds (vs 2.51 real, from above)
9-hit: 1.88 rounds (vs 2.32 real, from above)

If you could go from 16- to 15-hit easily, does that increase WS frequency by 6.39% (dummy 3.33/3.13) or by 5.65% (real 3.74/3.54)? Am I doing it right? Also, to compare to single swings 16- to 15-hit (no multi, no offhand), it should be a 6.67% increase in WS frequency, correct?

Or, to do the same with 10- to 9-hit:
Dummy: 2.08/1.88 = 10.638% increase in WS frequency
Real: 2.51/2.32 = 8.190% increase in WS frequency
Single: 10/9 = 11.111% increase in WS frequency

And finally, comparing 16- to 9-hit:
Dummy: 3.33/1.88 = 77.128% increase in WS frequency
Real: 3.74/2.32 = 61.207% increase in WS frequency
Single: 16/9 = 77.778% increase in WS frequency

If this is correct then STP has a slightly smaller benefit for KC than it does for a "normal" weapon, easiest to see in the 61% vs 78% high-STP example of course. That is what I wanted to know, and I would be fascinated as always to know, if I am doing it wrong, how and why. :)

Thank you math people!! :)

[CDF]

If this is correct then STP has a slightly smaller benefit for KC than it does for a "normal" weapon, easiest to see in the 61% vs 78% high-STP example of course. That is what I wanted to know, and I would be fascinated as always to know, if I am doing it wrong, how and why.

Thank you math people!!

As a check on your numbers, I repeated this exercise with a conditional expectation approach instead of probability tables (Markov chains) and got this for 100% hit rate:

http://img651.imageshack.us/img651/8337/kclub100hit.png

They seem close enough (I use the second column to check the extent of "TP overflow" and ensure the average number of hits/round is correct, 4.8 in this case). Could be rounding errors or incomplete description of attack round distributions with your approach.

Also did this for 95% hit rate:
http://img696.imageshack.us/img696/3761/kclub95hit.png

[RyaWHM]

Wow, thank you sir! That is so interesting! I even know what a Markwahlberg chain is! But no idea how to make it work in mathland, sadly :(

Yes I rounded a little, and merged some of the sub-half-percent outcomes -- but our numbers are very close EXCEPT ... lookit! Our numbers are off by one row. How did that happen?

Rya: 9, 10, 11, 12 hit build is 2.3, 2.5, 2.7, 2.9 rounds
CDF: 9, 10, 11, 12 hit build is 2.1, 2.3, 2.5, 2.7 rounds

But more importantly, can you tell me if my conclusion is fair? That STP has a smaller benefit for KC?

[CDF]

Oops, my "no. of hits to reach 100 TP" is the nominal number of hits to gain 100 TP from 0 TP, so subtract 1 to get the number of hits to reach 100 TP accounting for WS TP return.

Example: your 9-hit from WS to WS (2.32 attack rounds) corresponds to my 10-hit (9 hits from WS to WS, 2.418 ).

I would be willing to say that reducing the number of attack rounds to 100 TP has a lesser benefit for KC (and other multi-attack situations) ignoring the "costs" of store TP, and that the cost of using store TP to do so is much higher (both in a marginal return sense and opportunity cost) to get the appropriate TP per hit to go from a given n-hit to (n-1)-hit setup.