So, maybe it was just the magical EHP I chose, but from what I could tell, if the ADC has both 
and 
Your balance of hp and armor doesn't matter. Having 6000 hp and no bonus armor will kill you just as quickly as 2500 hp and 280 bonus armor (100 base armor in both cases).
I discovered this oddity by running dozens of simulations of stuff on R for a few hours.
R181 votesHe knows too much, he must be eliminated. But I'm pretty sure this works with multiple item combinations
TehNACHO1 votesJust checking, did you actually put this on an ADC (say, Tristana) and calculate including Tristana's base AD and all that, or entirely based on BotRK's passive?
Z Statistic1 votesI calculated it for 150 AD and 300 AD and played around with target dummy regen, armor, penetration, and uptime.
The measuring unit was in attacks (not time), so while I'm sure other ADC will get more use out of it purely from more AS, it'll take the same number of attacks on average.
If you're wondering what I didn't account for in my calculations, I didn't account for magical on-hit, % max and missing health, love taps, grievous wounds, and spells. You're just smacking away at fools like Ashe without using W or Q.
Or like Kindred. I was able to play around with Kindred's passive, though that's less realistic since she wouldn't be using QWE.
I think the big take away is that if the enemy ADC has BoRK+ LDR, you will tailor your defensive build around the other parts of the ADC and not around their basic attacks.
TehNACHO1 votesOkay let me test this real quick:
Damage per hit: (150+0.06HP)(100/(100+0.55*Armor))
6000 HP and no Armor (150+0.06*6000)=510 damage, about 12 shots to kill
2500 HP and 280 Armor (150+0.062500)(100/(100+0.55*280))=118 damage, about 22 shots to kill.
I don't get it.
That said, unless you're like Rammus or Malphite or looking at a Warden's Mail or Thornmail, the ADC is generally tailoring their build around their targets, not the other way around.
Z Statistic1 votesBoRK is current health, not max health. In order to solve for hits required, you have to solve the differential equation.
Or simulate it like I did. The plots I'm looking at are comparing the probability that it takes at most X hits since I'm letting uptime and crits be random variables.
Builds are tailored against each other. Defensive builds will generally be more sensitive to the enemy than offensive builds, but both respond to changes in the opponent's items.
TehNACHO1 votesRight, or you could just half the max health. The average if you're 100-0'ing somebody is 50% health after all.
Correcting that still comes to:
Damage per hit: (150+0.03HP)(100/(100+0.55*Armor))
6000 HP and no Armor (150+0.03*6000)=330 damage, about 19 shots to kill
2500 HP and 280 Armor (150+0.032500)(100/(100+0.55*280))=89 damage, about 29 shots to kill.
And could you just show us what you mean by simulate? Simple calculations that can show your work have a lot more value than "simulations" that only you apparently know since others can peer review and point out flaws, like you did mine, and show a more accurate model of the game. What is uptime supposed to be in this situation? Why are crits a variable when you already have a defined build? Why are you trying to sell this off as a one size fits all pattern of the game when you admit that Marksman like Kindred or Ashe skew your over controlled simulation so badly?
Furthermore, only Mail items actually directly respond to AD Threats.
Every time you make a new reply you're only confusing me further why you are turning such a simple thought experiment into such a convoluted mess. Everything else has primarily offensive applications that care far less about how the opponent is counter building you, or is Ice Born Gauntlet. Besides that, due to the mechanics of the game, it only makes sense that Marksman are countering Tanks.
Z Statistic1 votesNo need to get snarky. All you had to ask was for the source code.
https://github.com/z-statistic/League-Damage-Sim/blob/master/.gitignore The above code is to be use in R.
But to answer your questions:
Uptime is the % of opportunities you use to attack to attack. If you have 50% uptime, you are not attacking 50% of the time that you should be for whatever reason, maybe out of range or cced or misclick.
Crits are a random variable because crits are randomly decided? It can be useful to simulate the distribution instead of calculating the expected value.
I'm not.
You're confusing yourself. I'm only saying that BoRK and LDR together mean that the balance of hp/armor is irrelevant when considering EHP vs basic attacks and THEREFORE, you should consider the other parts of the ADC's kit or the other team's when making balance of EHP decisions.
It's to answer the question, "Shit, Vayne has both BoRK AND LDR. Should I build more armor or hp to counter her?" with "It doesn't matter. Build HP to absorb magic damage from Vayne's team or armor to counter her team's physical damage."
edit: you also did incorrect calculation.
Damage per hit: (150+0.03HP)(100/(100+0.55*Armor))
6000 HP and no Armor (150+0.03*6000)=330 damage, about 19 shots to kill
2500 HP and 280 Armor (150+0.032500)(100/(100+0.55*280))=89 damage, about 29 shots to kill.
It's 6000 with 100 base armor, and 2500 with 280 bonus armor and 100 bonus armor. So the calculations should be:
(150+0.036000)(100/(100+100)) = 165 6000/165 ~ 36
(150+0.032500)(100/(100+100+0.55*280)) = 63.55932 2500/63.55932 ~ 39
This is within the error you setup for yourself by estimating the solution to the differential equation by he linear function you created.
TehNACHO1 votesMac/Safari op.
And even then your conclusion is still wrong. In the first place, your 6000 HP no Armor and 2500 HP 280 Armor examples don't even have the same amount of EHP, 2500 HP and 280 armor is 9500 EHP to the 6000 EHP example. You didn't even balance your examples to each other, you can't say anything about balancing the individual Armor and HP of each. If you had, for example, 2000 HP and 200 Armor, which is the same EHP of 6000 HP and no Armor:
(150+0.032000)(100/(100+0.55200))=100 damage per hit, 20 hits to kill.
Look, I just found data that very nearly supports, but ultimately still disproves your claim, in such a far more simple manner. For the same amount of EHP, playing around with the ratio of HP vs Armor does in fact skew the number of shots it takes to kill the hypothetical target. It's not a matter of "is it relevant to itemize for the Auto Attackers Auto Attacks as well?", because it is relevant how you do so. Vayne should not be exempt in balancing your EHP against the enemy team, how you balance your items still has a definite effect on her effectiveness against you.
So to answer the question, should you build more armor or hp to counter her? My calculation concludes Armor, realistically you should still be building both and there are items to support both stats.
Much more realistically, this is still pretty jank given that Mail items are still better against Vayne than trying to fine tune and minmax the hell out of your stats.
Z Statistic1 votesYou might want to read things before you post responses. I very clearly state in the OP that the two target dummies were 6000 HP and 0 bonus armor with 100 base armor and 2500 HP with 280 bonus armor and 100 base armor. I'll let you confirm that they both indeed have 12k EHP.
And by the way: http://cran.us.r-project.org/bin/macosx/
If you don't want to bother checking my work (I actually did make a few mistakes in the sim, but I corrected them and ran it again for no real change), it is perfectly fine to say so. But don't get indignant and accuse me of hiding my work and then give up when the smallest barrier of entry pops up.