math stuff on HP & AC

[Phantron]

Using this parse http://members.roadfly.org/Frodlin/MitParse2.htm as our baseline. Using the AC = 3297 because it has the numbers that are easiest to compute. For sake of simplicity, getting hit for 18 DI = 1%, 19 DI = 1%, 20 DI = 0.5% and we normalize it to 2/2/1.

We're start being concerned with some extremes, and nothing's more extreme than hitting hit for 4DB + 80 DI (max every hit, 4 times). We'll also consider and 78 and 79 DI case.

chance of getting hit for DI 80 = (20/20/20/20) = 1
chance of getting hit for DI 79 = (20/20/20/19) X 4 combinations = 8
chance of getting hit for DI 78 = (20/20/20/18) X 4 + (20/20/19/19) X 6 = 8 + 24 = 32

So if your total HP is 4DB + 78 DI, your chance of dying in one round is 41*something. If your total HP is 4DB + 80 DI, your chance of dying in one round is 1*something. So, 2 DI of HPs reduces your chance of dying by about 97%. Even if DI = 200, that's only 400 HP.

So if AC can be considered exponential survival, so is HP.

Now we can consider tradeoffs. I claim 1AC = 3 HP according to this parse earlier, so if I lose 2 DI at DI = 200 of HPs I gain around 133 AC. Looking at the table, it seems downright unlikely the chance of getting hit by 18/19/20 DI can be reduced by half for any reasonable amount of AC we add, but we'll assume it's actually possible for only 133 AC. So we calculate this adjusted chance of survival with 4DB + 78 DI of HP. The math is simple. Since each hit is 1/2 as unlikely, the chance of getting hit for DI 78-80 is 1/16th of what it was before

chance of getting hit for DI 80 = 1/16
chance of getting hit for DI 79 = 8/16
chance of getting hit for DI 78 = 32/16
Total = ~2.5

The chance for the person with 4DB+80DI HP to die is still 1.

So how come we die more often even though we lowered the chance of a spike by 1/16th (including the spike for 78 and 79 DIs)? By losing 2 DI of HPs, you opened yourself to the chance of dying to the rounds of DI 78 and 79 that did not exist before, which occurs far more frequently than DI 80. Even if we assume the chance of getting hit for DI 18/19/20 are exactly the same (clearly parses show this is not what happens), you'd get 1 versus 1+4+10/16, or 1 vs 15/16. In this case the AC person will have a very slight edge in survival assuming all hits not on the modal are uniformly distributed.

It can be observed from parses that if you add enough AC, eventually DI 20 becomes the least common hit (which is good), and at that point, a mob will always hit X-1 DI easier than X DI for all DI values that matter. That is, it's always easier for a mob to hit a DI 19 than a DI 20, and easier still to hit a DI 18 than a DI 19. So, not only does the number of permutations for lower DI combinations increase (there are 1 way to hit DI 80, but 14 ways to hit DI 78), but because DI 20 is the rarest event, this means not only there are 14 ways to hit DI 78 versus 1 way for DI 80, each of those ways to hit DI 78 is more likely to occur than DI 80, as DI 80 requires 4 consecutive hits of the rarest event.

So once DI 20 is your rarest hit, the 'exponential' gain by AC is canceled out by the fact that when you lose HP, you can die to exponentially more common events as well.

[Phantron]

Continuing this, if max is not the most common hit, let's say the distribution of 18/19/20 DIs is 1/1/2, then you'd get:

4DB + 80 DI HP survival rating = 2^4 = 16
4DB + 78 DI HP + enough AC to reduce DI 18/19/20 DI distribution by half survival rating = 1^4 + 4 (0.5) + 4(0.5) + 6(0.25) = 6.5

The model of AC is necessarily inaccurate because we do not exactly understand how AC modifies each hit distribution, though I do not think this is an unreasonable simplification. How much AC is needed to achieve this is also irrelevant because it is constant even if we don't necessarily know what it is. By just changing the distirbution of hits on DI 18-20 we see very different results in favor of HP or AC. I do not have parses of ubers, but I feel that if DI 20 isn't the rarest hit, it certainly isn't more common than others. For the rest of this discussion I will use the assumption DI 20 = as rare as any other event. This means if the chance to get hit by 20 DI is 1, then every other DI value to get hit by is no less than 1 (besides the modal, which will be much higher). For the sake of easy computation (and this assumption is in favor of AC anyway), we'll assume all nonmodal values have a normalized chance to get hit of 1.

Will write more when raid is over...

Edit: High value on the survival rating means less likely to survive. This is probaly not intuitive...

[Kazak]

Nice analysis.

[Frodlin7th]

The problem is the math is completely wrong.

Let me explain...

Your chance at getting hit for DI = 20 is not additive, nor is it multiplied only once.

Let me explain in terms of flipping a coin. There's a 50% chance of getting heads on a single coin flip. There's a 25% chance of getting 2 heads on two coin flips. There's a 12.5% chance at getting 3 heads on three flips, and there's a 6.25% chance of getting 4 heads on 4 flips.

This is with an occurrence that happens with 2 possible numbers, each with equal possibilties of landing.

Now, if you take that same thing, and introduce 20 potential numbers, with the likelihood of one of the numbers landing not being "half" but less than 1%... more accurately, 0.55%

1 = 0.55%
2 = 0.27%
3 = 0.13%
4 = 0.06%

Essentially a 1 in 1440 chance at a full round of attacks at max

Using the same variable with the 2730AC value..

1 = 1.69%
2 = 0.72%
3= 0.36%
4= 0.18%

Or one in 552 chance for a full round to land for max

So you essentially almost triple your likelihood of preventing a full round of attacks by adding that AC. To turn that into 2 full rounds.... it would make it

At 3297AC a one in 23,040 chance

At 2730AC a one in 8,832 chance

There's actually a greater likelihood that someone with 2730AC will take two hits in a row for max damage than someone with 3297 will take a single one.

Now, I am a betting kind of guy, and I put my money where the bets are hedged for me in the most common kinds of death, which are... healer error, be it a late heal, or a lack of heal or whatver, I want to do what hedges my chances... I'm assuming that I'm maxed shielding/avoidance/Imp Dodge/Parry 3, because that I also want to wield to my advantage. The next best thing is to avoid the likelihood of being hit for max damage multiple times in a row, and that is ONLY found in AC.

[khur sed]

Actually, you can flip a coin a billion times in a row and always get head, and on he next flip, you still have a 50-50 chance of getting head yet again.

Do not confuse statistic with probability.

[Frodlin7th]

Yes, that's a completely different set of criteria. For each singular opportunity, the base chance of an event occuring is at the base chance. This isn't dealing with singular chances though, it's dealing with consecutive chances, and consecutive chances have a specific criteria by which they fall under, mainly how many potential combinations there are. So in your scenario, there's only two potential outcomes, head or tails. In a 2 flip combination there's 4 potential outcomes: Coin 1 heads, Coin 2 Tails... Coin 1 Tails, Coin 2 Heads, and the two doubles, thus the chance is 25% of getting the 1 in 4 combo.

With statistical analysis, a 0.55% chance represents 180 different potential combinations. A double consecutive would yield one in 360 potential combinations, the next would be 760, the next would be 1440. This is very very basic stuff, and the side issue about a single coin toss is really irrelevant to it.

[khur sed]

I'll bet ya pp to copper that SoE's curent dmg generator doesn't use common statistical distribution for their max and min hits.

You used an analogy with coins, who only have two possible outcomes. Yes, thats basic, and I was pointing out your analogy was wrong.

We must assume they are using sometype of formula that has to includes, but not limited to the following elements. Mob db, mob di, mob attack, mob attack speed, etc... then on the other side, the tank's ac, his shielding, his aa, his othr modifer, his level, and his buffs. So basically speaking, we're going on with at the very least, 10 different variable.

Are you two telling me you can make valid assumptions about 10 variable formula that you have in some way solved them?

Not to mention the extremely difficult problem of generating good enough random numbers with software.

Unless a dev is posting the exact math they are using, I'm weary of your mutual math.

It does make for interesting read, since you always find a way to disagree with one another.

[Vikken]

There's a fair bit of variance on those parses. Would all those parses be with the same AGI? The defense skills in particular are irregular enough to concern me. I know the main focus of the data is on mitigation, but I can't help but wonder.

BTW and isn't your example very mob specific. Mobs with different DI will have different points at which you are likely to be rounded by a slightly lower DI. In either case, it is far more likely that you won't even be hit for these values. At this point, doesn't healer effeciency matter more?

What I'm getting at, and not doing that great of a job of it, is that hitpoin vs AC ratio here is very mob specific. And now that I think about it, wouldn't leaning to the side of healer effeciency (lots o hitpoints) be better? Or are clerics already limited by CH caps?

[Egfrith]

What mob was used for those parses ?

[Yoda]

a testmob on test server with DB=172 and warrior's DI=30.3

[Sandaormo]

Me hit attack button , wait for cleric friend to heal. =)

[Elkar]

Frodlin's analogy is not incorrect, it is exactly correct. He is talking about the probablility of independent events occuring, prior to the event. It is also mathematically describing the benefit of high AC, in that it reduces the probability of a quad max round. Since it is a quad round in EQ, the value benefits from multiplicity to the fourth power.

Also, every parse I've seen shows the hit to be a very flat, but normal distribution, truncated for mins and maxes. That is, you will see a slight bulge in a graph right around the mean DI.

[Phantron]

Originally Posted by Frodlin7th

The problem is the math is completely wrong.

Let me explain...

Your chance at getting hit for DI = 20 is not additive, nor is it multiplied only once.

Let me explain in terms of flipping a coin. There's a 50% chance of getting heads on a single coin flip. There's a 25% chance of getting 2 heads on two coin flips. There's a 12.5% chance at getting 3 heads on three flips, and there's a 6.25% chance of getting 4 heads on 4 flips.

This is with an occurrence that happens with 2 possible numbers, each with equal possibilties of landing.

Now, if you take that same thing, and introduce 20 potential numbers, with the likelihood of one of the numbers landing not being "half" but less than 1%... more accurately, 0.55%

1 = 0.55%
2 = 0.27%
3 = 0.13%
4 = 0.06%

Essentially a 1 in 1440 chance at a full round of attacks at max

Using the same variable with the 2730AC value..

1 = 1.69%
2 = 0.72%
3= 0.36%
4= 0.18%

Or one in 552 chance for a full round to land for max

So you essentially almost triple your likelihood of preventing a full round of attacks by adding that AC. To turn that into 2 full rounds.... it would make it

At 3297AC a one in 23,040 chance

At 2730AC a one in 8,832 chance

There's actually a greater likelihood that someone with 2730AC will take two hits in a row for max damage than someone with 3297 will take a single one.

Now, I am a betting kind of guy, and I put my money where the bets are hedged for me in the most common kinds of death, which are... healer error, be it a late heal, or a lack of heal or whatver, I want to do what hedges my chances... I'm assuming that I'm maxed shielding/avoidance/Imp Dodge/Parry 3, because that I also want to wield to my advantage. The next best thing is to avoid the likelihood of being hit for max damage multiple times in a row, and that is ONLY found in AC.

You should stop using math because you obviously have no idea what it is.

If your chance of getting hit for DI 20 once is 0.55% then the chance of getting hit twice is 0.55%^2 = (0.0055^2) = 0.00003025. The chance of getting hit for DI 20 4 times in a row is (0.0055^4) which is roughly 1 in 1.1 billion. It sure would be nice if the chance of getting 2 events in a row is just event 1/2, but that's not how math works.

I purposely didn't use actual numbers and normalized them to 1 because it illustrates the absurdity of surviving the spikes in the first place. I'm sure you're now more confident staying alive because your chance of getting hit for 80 DIs in one bad round is 1 in 1.1 billion as opposed to (0.016^4) = 1 in 15 million at the other AC level you picked. After all the likelihood of such an event happening is about 1000 times rarer. Of course you completely ignore these aren't any event you'd expect to observe in all your entire EQ playing time in the first place, and also the fact that opening yourself up to just 2 more DI of HP increases the chance of dying by a factor of at least 14 (there are 14 ways to get hit for DI 79 and 78, and both are at least as likely as DI 80). Note that this is just for 2 DI of HP here. For the difference in the mitigation on the 2 AC level (~600 AC) you'd have to give up a lot more than 2 DI of HPs.

I went back and looked at your parse again. Givein DI = 200, adding 133 AC (at 1AC = 3 HP), it might be *possible* to actually lower the DI = 20 distribution by half (depending on if the 3297 AC data is an anaomly or not), but looking at the distribution it doesn't look like anything less than 1000 AC would get the DI = 18 or 19 distribution by half. So my assumption that you move a normalized distribution of 2/2/1 to 1/1/0.5 via AC is still grossly in favor of AC. The way you look at it, there is apparently only one spike value that you care about (DI = 80) and you completely ignore the fact that trading HP for AC means you can die to a lower combination of DI.

[Sabotage]

Originally Posted by Elkar

Frodlin's analogy is not incorrect, it is exactly correct.

You can't say that without knowing if the distribution of DI values are given the same probability of occuring on each round of damage through natural randomness or through some other type of system. SoE very well could have implemented an increased series of streaks into the system by design that would still promote the illusion of random chance. By doing so any parsings done by players would still result in the same perception of randomness, without the realization that a system was implemented to have increased damage spikes as well as increased periods of lacking-damage. This would allow them to throw a wrench in conventional healing by providing for a greater chance of extremes, while still keeping the basic foundation of probability to appear to be evenly spread out.

With the exception of small variables in tossing a quarter up such as wind velocity, surface area deformities and other similar things, you're pretty much 50/50 of flipping a quarter in the air and it landing on heads. However, such a system's randomness can be compromised by producing an increased chance of reproducing the same action through series of 3 or 4.

Take a system generated through programming, in which the code writer is looking to produce a higher number of streaks while keeping the overall randomness of each chance the same: Heads is flipped. On the 2nd flip, the probability of heads to tails is increased by the system to 60/40. On the 3rd flip the probability of heads to tails is increaed by the system to 70/30. On the 4th flip the probability of heads to tails is increaed to 80/20. If you produce the same results with the other option, in this case, tails, a large parsing of the data would produce the same # of heads/tails flips but further (and much more intensive) investigation into streakiness would provide surprising results.

I don't think anyone has fully investigated the possibility of hits occuring in streaky lumps in EQ - if they do occur then that's an argument that would benefit the HP argument as AC would be handicapped by the system in providing diminished returns on each subsequent hit of the streak.

People make way too many assumptions about how the system was developed, with very limited information at times.

[Elkar]

Originally Posted by Sabotage

You can't say that without knowing if the distribution of DI values are given the same probability of occuring on each round of damage through natural randomness or through some other type of system. SoE very well could have implemented an increased series of streaks into the system by design that would still promote the illusion of random chance. By doing so any parsings done by players would still result in the same perception of randomness, without the realization that a system was implemented to have increased damage spikes as well as increased periods of lacking-damage. This would allow them to throw a wrench in conventional healing by providing for a greater chance of extremes, while still keeping the basic foundation of probability to appear to be evenly spread out.

Good point, however if AC was accounted for in the streak, it would still hold the same value, maybe even moreso.