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## Left Eye of Rajh and Grace of the Herald

Face-rippin fun.

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### Left Eye of Rajh and Grace of the Herald

Good evening. I wanted to check where Left Eye of Rajh and Grace of the Herald currently are on the list for me.

Grace of the Herald
285 agility, 1710 crit on proc
Buff duration: 10s
ICD: 100s

(1710*0.751905)*((10s/100s)0.1)=12.8575755

285*2.58653=737.16105

12.8575755+737.16105=750.0186255

Left Eye of Rajh
171 expertise, 114 mastery, 1710 agility on proc
Buff duration: 10s
ICD: 50s

(1710*2.58653)*((10s/50s)0.3)=265.377978

(171*0.764472)+(114*0.900169)=233.343978

265.377978+233.343978=498.721956

Before you say anything - I know this is plain wrong. I get that Grace of the Herald has a total uptime of 1%, while I get that Left Eye of Rajh has a total uptime of 6%, both which are way too low. I read that Grace of the Herald's proc chance is 10% and that Left Eye of Rajh's proc chance is 30%. Help would be very much appreciated. Here are the relative stat values for my toon should you need them:
Code: Select all
__Stat__          __Value__      __Error__Weapon DPS        4.410591       0.037448Agility           2.58653        0.037173Strength          2.021381       0.037129Attack Power      0.964411       0.01856Mastery rating    0.900169       0.036978Hit rating        0.768199       0.036922Expertise rating  0.764472       0.036925Crit rating       0.751905       0.036986Haste rating      0.417530       0.036925

Also, does anyone know if either of the trinkets have changed anything with patch 4.0.6? Someone has reported that he experienced an improved proc chance on Grace of the Herald after the patch. Might be worth checking out.

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### Re: Left Eye of Rajh and Grace of the Herald

It wouldn't surprise me if Grace of the Herald was fixed to have a 50s ICD (that would be consistent with other 10s trinkets).

Let's do some rough math:
The Eye has a 50% proc chance (up from 30%) on crit. Assuming 40% crit rate, .4*.5 = .2, so 20% chance to proc on hit. Assuming an attack speed of .8, it should proc on average 2s after the ICD comes up. It can proc off specials as well, so it'll be less than that, but we'll leave that alone for now.
Since Grace has a 10% proc chance on hit, just double the above, and assume it procs on average 4s after the ICD comes up. That gives Eye a 19.23% uptime and Grace an 18.51% uptime. Simply multiply the uptime with the proc stat bonus to determine what the approximate static value is, and then use the relative stat values to calculate overall value.

Note: As you can see, proc chance doesn't affect uptime a whole lot for us (which is why I just ignored the effects of proc chance in my original trinket rankings).

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### Re: Left Eye of Rajh and Grace of the Herald

Was the Eye's proc chance increased with the patch? I haven't seen anything about it anywhere. And yes, it would make sense if Grace's ICD was changed to 50s, since it was after all the only trinket deviating from the ICD=buff duration*5 rule.

I see that your way of calculating it is different, but it makes somewhat more sense. I didn't take attack speed or crit chance into consideration (but that is mostly due to the fact that I didn't get that far, since I had it wrong anyway). The error of mine is most likely where I multiplied proc chance with 'raw' uptime (i.e. uptime before proc chance is taken into consideration), since I assumed proc chance was another way of interpreting uptime in general, but this depends of course on attack speed.

One question. How do you get 2s in average after the ICD comes off? I do agree that the chance for it to proc on hit is 20%. If the attack speed is 0.8 (i.e. one attack every 0.8 second), wouldn't it be 0.8*5=4s?

Alaron wrote:Simply multiply the uptime with the proc stat bonus to determine what the approximate static value is, and then use the relative stat values to calculate overall value.

Aye, that's what I did in my calculations in the first post, but with wrong static values. :p

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### Re: Left Eye of Rajh and Grace of the Herald

Drezleith wrote:Was the Eye's proc chance increased with the patch? I haven't seen anything about it anywhere. And yes, it would make sense if Grace's ICD was changed to 50s, since it was after all the only trinket deviating from the ICD=buff duration*5 rule.
...
One question. How do you get 2s in average after the ICD comes off? I do agree that the chance for it to proc on hit is 20%. If the attack speed is 0.8 (i.e. one attack every 0.8 second), wouldn't it be 0.8*5=4s?

Eye change- It was in patch notes, but without numbers (just "increased"). Wowhead had the number.

For average time to proc, we're actually both wrong; we both used common (incorrect) shortcuts. To get the exact answer, you need to use a binomial distribution (which I'm not doing right now). As an approximation, you want to know how many attacks n will occur to achieve 50% proc chance. If we say that 0.8 is the chance to not proc, then
1n: .8 chance to not proc
2n: .8*.8 = .64 chance to not proc
3n: .8*.8*.8 = .51 chance to not proc
4n: .8^4 = .41 chance to not proc

So, it should actually be a touch above 3 attacks, or about 2.5s. (If you think about it, you should understand why you're trying to reach 50% if you're looking for an average.)

Using similar math, Grace comes in at between 6 and 7 attacks, so it should take about 5-5.2s to proc it.

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### Re: Left Eye of Rajh and Grace of the Herald

Alaron wrote:Eye change- It was in patch notes, but without numbers (just "increased"). Wowhead had the number.

Where exactly? I searched around in the patch notes and all over the blog posts, but there wasn't anything there.

Alaron wrote:For average time to proc, we're actually both wrong; we both used common (incorrect) shortcuts. To get the exact answer, you need to use a binomial distribution (which I'm not doing right now). As an approximation, you want to know how many attacks n will occur to achieve 50% proc chance. If we say that 0.8 is the chance to not proc, then
1n: .8 chance to not proc
2n: .8*.8 = .64 chance to not proc
3n: .8*.8*.8 = .51 chance to not proc
4n: .8^4 = .41 chance to not proc

So, it should actually be a touch above 3 attacks, or about 2.5s. (If you think about it, you should understand why you're trying to reach 50% if you're looking for an average.)

Using similar math, Grace comes in at between 6 and 7 attacks, so it should take about 5-5.2s to proc it.

I was thinking exactly the same. I sat calculating the proc chance of each attack using binomial distribution, but I stumbled upon something that didn't make sense in this situation, so I deleted the suggestion from my post before submitting it.

If we go with proc chance each hit.
1st hit: 0.2 chance to proc
2nd hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{2%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^1%20=%200.32[/img]
3rd hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{3%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^2%20=%200.384[/img]
4th hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{4%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^3%20=%200.4096[/img]
5th hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{5%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^4%20=%200.4096[/img]
6th hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{6%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^5%20=%200.393216[/img]
7th hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{7%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^6%20=%200.3670016[/img]
...
100th hit: [img]http://www.matematikk.net/cgi-bin/mimetex.cgi?{100%20\choose%201}%20\cdot%200.2^1%20\cdot%20(1-0.2)^99%20=%205.092589941%20\cdot%2010^{-9}[/img]

As you can see the probability is declining by every hit after the 5th. In binomial distribution you want to know the probability for a given amount of independent success/failure experiments. We, however, want to know how many independent success/failure experiments it will take to yield the highest probability possible. That's the single reason it is wrong to use binomial distribution.

There is however a far easier way of doing this, which is actually the one I mentioned in my previous post. We know that the proc chance for each hit is 20%. This translates to 0.2 in Probability World, which again translates to 1/5. One in five attacks will proc the trinket effect. Another way of saying it is that, over the course of time, every fifth attack will proc the trinket effect, in average. We also get this if we divide 1 by 0.2. If we make that into seconds we get 4 ((1/0.2)0.8=4).

Edit: Seems that the img tags are not working. Just copy-paste into the URL field.

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### Re: Left Eye of Rajh and Grace of the Herald

Here's where I remember the patch note from: http://www.mmo-champion.com/threads/829372-Patch-4.0.6-Notes-Update-January-11

I am definitely getting far afield here, but when doing the binomial distribution, you're not looking for the chance that n = 1, you're looking for the chance that n >= 1. That's why your probability is declining with more trials. (And also why 5 trials is incorrect. If we were looking for n=1, 5 trials would be correct.)

Technically, there is no "average" time to proc, as the % of a proc approaches 1 but never reaches it. Hence, my definition of "average" is "the point in time where 50% of proc times fall before it, and 50% of proc times fall after it." I'm sure someone better versed in statistics could give a better, more exact answer.

Try this calculator: http://faculty.vassar.edu/lowry/binomialX.html

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### Re: Left Eye of Rajh and Grace of the Herald

Alaron wrote:Here's where I remember the patch note from: http://www.mmo-champion.com/threads/829372-Patch-4.0.6-Notes-Update-January-11

I must be blind, since I can't see where it states that the Eye's proc chance was increased frrom 30% to 50%.

Alaron wrote:I am definitely getting far afield here, but when doing the binomial distribution, you're not looking for the chance that n = 1, you're looking for the chance that n >= 1. That's why your probability is declining with more trials. (And also why 5 trials is incorrect. If we were looking for n=1, 5 trials would be correct.)

I know, because there is very litte chance to get exactly one in, say, one hundred. My point was simply that we were looking for how many attacks (analogically the independent success/failure attempts), not the probability of a given amount of attempts.

Alaron wrote:Technically, there is no "average" time to proc, as the % of a proc approaches 1 but never reaches it. Hence, my definition of "average" is "the point in time where 50% of proc times fall before it, and 50% of proc times fall after it." I'm sure someone better versed in statistics could give a better, more exact answer.

I do understand your explanation since I have experience on this subject as well.

Heck, I'll just run some tests in-game with this, so we can get a definite answer. I thank you very much for your input, though.

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### Re: Left Eye of Rajh and Grace of the Herald

I ran a few tests in-game to learn how long time it takes for Grace of the Herald and Left Eye of Rajh to proc.

I had no buffs, one trinket equipped at a time and only auto-attack going (including Fury Swipes). I used the stopwatch in-game (which I cursed occasionally for not displaying desimals) and the stopwatch on my cellphone, alternating between them to make it more accurate, and before I started the tests I ensured that both where running at the same speed, as some clocks/stopwatches run faster or slower than others.

Also, the reason there are separate lines under each trinket is that there were mistakes, so I ran out and started over after the ICD was off again. 20 procs on each trinket.

Grace of the Herald

Relevant information
- Buff duration: 10s
- Presumed ICD: 100s/1m 40s
- Presumed proc chance: 10%
- Attack speed during tests: 0.94

3s, 1m 42.56s, 1m 51s, 1m 45.78s, 1m 49s, 1m 41.32s, 1m 40s, 1m 44.92s, 1m 47s, 1m 40.75s (4.533s)

5.61s, 1m 45s, 1m 45s, 1m 40.39s, 1m 42s, 1m 43.06s, 1m 42s, 1m 42.37s (3.17875s)

10s

1.53s

Average proc time: 4.1145s
Proc chance: 22.84603232%

Left Eye of Rajh

Relevant information
- Buff duration: 10s
- ICD: 50s
- Proc chance: >30%
- Crit chance during tests: 35.60%
- Attack speed during tests: 0.94

2.57s, 51s, 50.33s, 1m 6s, 54.84s, 56s, 51.87s (4.658571429s)

2.03s, 1m 6s, 55.94s, 53s, 53.17s (6.028s)

5.06s, 1m 1s, 50.54s, 53s, 57.52s, 55s, 1m 0.39s, 1m (6.56375s)

Average proc time: 5.763s
Proc chance: 45.81727292%

While someone reported that the ICD of Grace of the Herald was possibly lowered with the recent patch, this confirms that the ICD is still at 100s, but the proc chance seems to have been increased. As for Left Eye of Rajh the proc chance has indeed increased, as was stated in the patch notes.

Although, the variation was big, I would guess that Grace of the Herald's exact proc chance is 20% now, while Left Eye of Rajh's exact proc chance is 50%. The reason for this is simply because there are a very small amount of incidents on Grace of the Herald where it procced right off the ICD, whereas most of the procs used several seconds to trigger. On Left Eye of Rajh's part it's the opposite; there are a small amount of procs that used more than 10 seconds to trigger, whereas most of the procs were around 5 seconds.

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### Re: Left Eye of Rajh and Grace of the Herald

Awesome work, Drez. Real on-the-ground observation is always better than math.

Looking at your numbers, it looks like they're higher than we both figured, which is interesting, and makes we want to test some of the higher trinkets. We probably need to figure out a better way to do testing though (as much as I respect your stopwatch skills). There used to be an addon (possibly called Proculas?) that helped with this kind of thing.

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### Re: Left Eye of Rajh and Grace of the Herald

Thank you very much. :3 It indeed is; things never really turn out exactly how they are on paper.

I was thinking the same, as I felt somewhat stupid sitting there with stopwatches. Though, with a lot of concentration, knowing about when it was about to proc and staring on the buffs pane at that time, I think I got it pretty accurate. Would of be relieving with an accurate tool, specifically designed for this. I will search around and see if I can find anything matching this.

Keep in mind that this was only auto-attacks, so in reality it would proc faster in average.