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cheese_cake
02.19.2020 , 06:53 PM | #2
In F1 power setting for both shields and hulls:
Shield piercing is accounted for in this version. Pile Driving with the ion build is much more thoroughly considered too.

TL;DR
Again at all ranges the ions build has a higher theoretical probability to win the joust. At extreme close ranges the difference is marginal. At extreme close ranges the quads build can utilise a rotation with a marginally lower chance to eliminate its target with an approx. .23s lower time to kill.

Both ships:
1525 hull points
12.5% evasion
Shield pool of 2104.2 in F1
Accuracy and damage are assumed to scale linearly with distance.

Ion Cannons:
@range 5750m
Let the event that Ions hit their target be P(I)
P(I) = 89.5%
Damage per shot to shields ≈ 492.96 * 1.25 ≈ 616.2
@range 525m
P(I) ≈ 109.37%
Damage per shot to shields ≈ 696.34 * 1.25 ≈ 870.425

Quad Lasers:
@range 5750m
Let the event that Quads will hit their target be P(Q)
P(Q) ≈ 79.69%
Per shot shield ≈ 301.55 * 1.25 ≈ 376.9375
Per shot hull ≈ 446.45
@range 525m
P(Q) = 103.5%
Per shot shield = 390 * 1.25 = 487.5
Per shot hull ≈ 577.31

Heavy Lasers:
@range 5750m
Let the event that Heavies will hit their target be P(H)
P(H) ≈ 90.98%
Per shot shield ≈ 436.65 * 1.25 ≈ 545.8125
Per shot hull ≈ 589.56
@range 525m
P(H) = 111 – 12.5 = 98.5%
Per shot shield = 503 * 1.25 = 628.75
Per shot hull = 679.375

Calculating probabilities:

@range 5750m

Pile driving stage of ion build to strip shields:
Second heavy shot with an overflow of 219.825 damage to hulls. This takes shield penetration from heavies into account.
Let the event that all shots up to the second heavy shot hits & remove shields for the ion build be P(Si).
P(Si) = P(H ∩ I ∩ I ∩ H) = P(H)P(I)P(I)P(H) = .895^2 * .9098^2 ≈ 66.30%

Pile driving stage of the quads build to strip shields:
Third quad shot with an overflow of 118.2375 damage.
Let the event that all shots up to the third quad shot hits & remove shields for the quads build be P(Sq).
P(Si) = P(H ∩ Q ∩ Q ∩ H ∩ Q) = P(H)P(Q)P(Q)P(H)P(Q) = .7969^3 * .9098^2 ≈ 41.89%

Shots required by the ion build to eliminate hulls:
3 heavy shots.
Let the event that all shots required by the ion build to eliminate hulls be P(Hull i).
P(Hull i) = P(H ∩ H ∩ H) = P(H)P(H)P(H) = .9098^3 ≈ 75.31%

Pile driving stage by the quads build required to eliminate hulls:
Fourth heavy shot.
Let the event that all shots after the third quad shot to the fourth Heavy shot hits from the quads build to eliminate hulls be P(Hull q).
P(Hull q) = P(H ∩ Q ∩ H) = P(H)P(Q)P(H) = .9098^2 * .7969 ≈ 65.96%

P(Ei) = P(Si)P(Hull i) = .6630 * .7531 ≈ 49.93%
P(Eq) = P(Sq)P(Hull q) = .4189 * .6596 ≈ 27.63%

Timelines:
Ion build:
0s – First heavy shot & swap.
.5s – Second ion shot & swap.
.5s – Second heavy shot. Shields stripped.
2s – Fifth heavy shot. Target eliminated.

Quads build:
0s – First heavy shot.
.74s – Third quad shot. Shields stripped.
1.57s – Fourth heavy shot on swap. Target eliminated.

The cooldown of the systems ability is .5s. The quads build will not be able to squeeze in another shot before the 2s mark. In the scenario where both ships are in F1 the entire duration at maximum ion range, the ion build is about twice as likely to eliminate its target.

If instead the quads build waits till the 1.11s mark after the third quad shot to do its HQQH combo, it can theoretically finish its target by the 1.61s mark. It will need either HQH or HQQ to finish its target.

The alternative P(Hull q):
P(Hull q) = P(((H ∩H) ∩(Q ∪ Q))∪((Q ∩Q)∩(H ∪H)))
P((H ∩H) ∩(Q ∪ Q)) = P(H)P(H) * (P(Q) + P(Q) – P(Q)P(Q)) ≈ 79.36%
P((Q ∩Q)∩(H ∪H)) = P(Q)P(Q) * (P(H) + P(H) – P(H)P(H)) ≈ 62.99%
P(Hull q) ≈ .7936 + .6299 - .7936 * .6299 ≈ 92.36%

Alternative P(Eq) ≈ .4189 * .9236 ≈ 38.69%

The alternative is mutually exclusive from the first iteration i.e. both cannot happen at the same time. While the alternative bumps the probability of the quads build to win by about an additional 10%, it is still about 10% behind the ion build. Either the quads build eliminates its target within the <2s window, or it does not.

@range 525

Pile driving on ions for total target elimination:
The ion build can strip shields within the .5s window with either two ions then one heavy (265.4 damage to hull) or one heavy then two ions (94.3125 damage to hull). Shield piercing accounted for. The time to kill for both events are the same, but the latter event requires 3 heavy shots to connect on hull whereas the former requires two. Because of this the calculation will consider the union of both entire events.

Let the event of the IIH ion option for full target elimination be P(Ei1) and the HII option be P(Ei2).
P(Ei1) = P(I ∩ I ∩H ∩ H ∩H)) = P(I)P(I)P(H)P(H)P(H)
P(Ei2) = P(H ∩ I ∩ I ∩H ∩ H ∩H)) = P(H)P(I)P(I)P(H)P(H)P(H)

Ergo,
P(Ei) = P(Ei1 ∪ Ei2)

Where P(I) = 1
P(Ei) = P(H)P(H)P(H) + P(H)P(H)P(H)P(H) - P(H)P(H)P(H)P(H)P(H)P(H)P(H) ≈ 99.74%

Pile driving stage of the quads build to strip shields:
Second heavy shot with an overflow of 128.3 damage.
Let the event that all shots up to the second shot shot hits & remove shields for the quads build be P(Sq).
P(Sq) = P(H ∩ Q ∩ Q ∩ H) = P(H)P(Q)P(Q)P(H)

Pile driving stage by the quads build required to eliminate hulls:
Fourth quad shot
Let the event that all shots after the second heavy shot to the fourth quad shot hits from the quads build to eliminate hulls be P(Hull q).
P(Hull q) = P(Q ∩ H ∩ Q) = P(Q)P(H)P(Q)

Where P(Q) = 1,
P(Eq) = P(H)P(H)P(H) ≈ 95.57%

Alternatively, the quads build can wait .5s to do its HQQH combo again. This will align its Time to Kill with the ion build. In this case the quads build needs to nail one of HQQ, QQH, or HQH.

P(Hull q alternative) = P(HQQ ∪ QQH ∪ HQH)
= P(HQQ) + P(QQH) + P(HQH) – P(HQQ)P(QQH) – P(HQQ)P(HQH) – P(QQH)P(HQH) + P(HQQ)P(QQH)P(HQH)

Where P(Q) = 1,
P(Hull q alternative) = P(H) + P(H) +P(H)P(H) – P(H)P(H) – P(H)P(H)P(H) – P(H)P(H)P(H) + P(H)P(H)P(H)P(H)
= 2*P(H) – 2*P(H)^3 + P(H)^4
≈ 100%

P(Eq alternative) ≈ 97.02% (Equal TTK to ions)
P(Eq) stated again ≈ 95.57% (shorter TTK than ions)
P(Ei) stated again: ≈ 99.74%

Timelines:

Ion:
0s –First heavy & ion shot
.5s – Second ion shot & second heavy shot. Shields stripped.
1.5s – Fourth total possible heavy shot. Target eliminated.

Quads:
0s – First heavy shot
.5s – Second heavy shot. Shields stripped.
Either:
1.27s – Fourth quad shot. Target eliminated.
OR:
1s – Wait till combo resets and fire new first heavy shot.
1.5s – One of HQQ, QQH, or HQH. Target eliminated.