Enhancement probability guide for rares (with onyx)

[Cygnus]

*these data are done with my own calculations and the DNArmory enhancement guide*

*This guide is only for rare equips*

This list contains the probability of enhancing an item all the way to +15 including the amount of onyx needed. The data are all theoretical mathematical calculations. Real world results are needed and appreciated from you to confirm if my numbers are accurate.

*Please only use this guide as a reference and an idea of how much onyx you will need*

Factors and definitions:
The degrading rate and the breaking rate are calculated after success rate, not as a total 100%, this would mean if the breaking rate is 25% at level 7, it is 25% of the 65% which totals it up to 13.75%

I have used the term static rate to define the possibility of either only failing or succeeding, it does not include breaking or degrading

for the onyx calculation, it only factors in passing, failing and degrading, not breaking



feedback and real world data are appreciated, thank you

[JTRedstar]

Nice Guide !

Wondering if this guide is also the same in SEA server, is it ?

[Vagitarian]

Thank you for this!

[crazytrout]

Thanks a bunch for this. Always hate failing my enhances -__-

[Teddy]

Not sure how I'm supposed to read this thing but it looks useful :|

[KholdStare88]

Necromancing, but I think I have a helpful post regarding enhancement chances. In this post we will try to find the probability of starting at lvl 0 (clean) and obtaining level of your choice, which may be interesting for some of you to read.

Let us define the following variables

p# = probability of going from lvl 0 to lvl #
i# = probability of going from lvl (# - 1) to lvl #
r# = eventual rate of success from lvl (# - 1) to lvl # without decreasing
d# = eventual rate of failure and decreasing from lvl (# - 1) to lvl (# - 2)

When I refer to "raw", that means the values given on Cygnus' table. Now I will explain what these variables mean and how they differ from the raw values listed in the table.

p8: this is what we're trying to find, the probability of achieving level 8 from lvl 0 without breaking
i7: this is assuming if you already achieved level 6, what is the probability you will eventually after unlimited tries get to level 7 without breaking, and it is simply p7/p6 (note this probability accounts for decreasing as many times as you want)
r8: assuming we have unlimited trials, the fail rate while staying at the same level doesn't matter since we will repeat the trial, hence r8 = .4/(.4+.21+.15), using values directly from Cygnus' table
d8: similar to r8, we define d8 = .21/(.4+.21+.15)

We are ready to start our analysis. Without much surprise to anyone, we have:

p1 = 100%
p2 = 100%
p3 = 100%
p4 = 100%
p5 = 100%
p6 = 100%

If there is no chance of breaking, then naturally with unlimited trials you will eventually achieve success. In other words, you can get to lvl 6 without breaking and everybody knows this.
______________________________

Now we consider p7. Since your item cannot decrease while getting to lvl 7, we don't have that to worry about. This means we only have to worry about raw success vs. breaking chance. Hence:



So starting with a clean rare item, the probability you'll get it to lvl 7 is about 3/4 of of time, which means on average you'll break 1 out of 4.

For p8, it becomes more interesting. We will have to break this into parts.

a) First, let's assume no decreasing or breaking. Then the probability of getting from lvl 7 to lvl 8 is simply p7r8, where we multiply the probability of successfully achieving lvl 7 by the eventual rate of (7 -> 8).

b) Second, we consider the case were we decrease once then success, or (7 -> 6 -> 7 -> 8). This probability is d8i7 * p7r8, and it's not hard to understand. The d8 accounts for decreasing to lvl 6, and the i7 accounts for going back to lvl 7. Then we multiply because of "AND" by the probability in part (a), which is (7 -> 8).

c) But that's not all, since it can go (7 -> 6 -> 7 -> 6 -> 7 -> 8), and in fact it can decrease to 6 as many times as it wants. Therefore, we will need to use an infinite series to represent p8, which is the sum of all these discrete scenarios. In other words:



Recall that when you decrease to lvl 6 twice, you square the d8i7 term, but you leave the p7r8 alone since you will decrease twice, get back to lvl 7 twice, but get to lvl 8 only once. And we will add on all the other cases. Now, we can factor out p7r8 from this equation:



And using the formula for convergence of infinite series, we obtain:


______________________________

For p9, it is very similar with one difference. The astute reader will notice that in the previous example, i7 = p7/p6, but p6 is 100%, so in fact i7 = p7. Now that's no longer the case, because now i8 will also account for decreasing from 8 -> 7 -> 6 -> (then all variations with finally) -> 9. So therefore, we have:



And we can now generalize this formula to:



Note that i# depends the p# and p(# - 1), which means we have a recursive relation that depends on the previous two terms. Applying the formula for the rest, we find the remaining probabilities to be:

p7 = 76.6%
p8 = 51.1%
p9 = 30.4%
p10 = 15.9%
p11 = 7.06%
p12 = 2.54%

So in other words, starting with a clean piece of rare gear, we have a 2.54% chance of enhacing it to lvl 12, or approximately 1 in 39 items will enhance successfully to lvl 12. If you're really interested, here are the probabilities for higher levels:

p13 = 0.690%
p14 = 0.0651%
p15 = 0.0000114%

I wouldn't trust these numbers though, because I can't imagine how Cygnus obtained these values without sufficient testing. Thanks for reading, and if you're interested, I will do another post on how unlucky you really are when you break 5 items in a row trying to enhance to +8, etc...

[profdharyl1487]

Originally Posted by KholdStare88

Necromancing, but I think I have a helpful post regarding enhancement chances. In this post we will try to find the probability of starting at lvl 0 (clean) and obtaining level of your choice, which may be interesting for some of you to read.

Let us define the following variables

p# = probability of going from lvl 0 to lvl #
i# = probability of going from lvl (# - 1) to lvl #
r# = eventual rate of success from lvl (# - 1) to lvl # without decreasing
d# = eventual rate of failure and decreasing from lvl (# - 1) to lvl (# - 2)

When I refer to "raw", that means the values given on Cygnus' table. Now I will explain what these variables mean and how they differ from the raw values listed in the table.

p8: this is what we're trying to find, the probability of achieving level 8 from lvl 0 without breaking
i7: this is assuming if you already achieved level 6, what is the probability you will eventually after unlimited tries get to level 7 without breaking, and it is simply p7/p6 (note this probability accounts for decreasing as many times as you want)
r8: assuming we have unlimited trials, the fail rate while staying at the same level doesn't matter since we will repeat the trial, hence r8 = .4/(.4+.21+.15), using values directly from Cygnus' table
d8: similar to r8, we define d8 = .21/(.4+.21+.15)

We are ready to start our analysis. Without much surprise to anyone, we have:

p1 = 100%
p2 = 100%
p3 = 100%
p4 = 100%
p5 = 100%
p6 = 100%

If there is no chance of breaking, then naturally with unlimited trials you will eventually achieve success. In other words, you can get to lvl 6 without breaking and everybody knows this.
______________________________

Now we consider p7. Since your item cannot decrease while getting to lvl 7, we don't have that to worry about. This means we only have to worry about raw success vs. breaking chance. Hence:



So starting with a clean rare item, the probability you'll get it to lvl 7 is about 3/4 of of time, which means on average you'll break 1 out of 4.

For p8, it becomes more interesting. We will have to break this into parts.

a) First, let's assume no decreasing or breaking. Then the probability of getting from lvl 7 to lvl 8 is simply p7r8, where we multiply the probability of successfully achieving lvl 7 by the eventual rate of (7 -> 8).

b) Second, we consider the case were we decrease once then success, or (7 -> 6 -> 7 -> 8). This probability is d8i7 * p7r8, and it's not hard to understand. The d8 accounts for decreasing to lvl 6, and the i7 accounts for going back to lvl 7. Then we multiply because of "AND" by the probability in part (a), which is (7 -> 8).

c) But that's not all, since it can go (7 -> 6 -> 7 -> 6 -> 7 -> 8), and in fact it can decrease to 6 as many times as it wants. Therefore, we will need to use an infinite series to represent p8, which is the sum of all these discrete scenarios. In other words:



Recall that when you decrease to lvl 6 twice, you square the d8i7 term, but you leave the p7r8 alone since you will decrease twice, get back to lvl 7 twice, but get to lvl 8 only once. And we will add on all the other cases. Now, we can factor out p7r8 from this equation:



And using the formula for convergence of infinite series, we obtain:


______________________________

For p9, it is very similar with one difference. The astute reader will notice that in the previous example, i7 = p7/p6, but p6 is 100%, so in fact i7 = p7. Now that's no longer the case, because now i8 will also account for decreasing from 8 -> 7 -> 6 -> (then all variations with finally) -> 9. So therefore, we have:



And we can now generalize this formula to:



Note that i# depends the p# and p(# - 1), which means we have a recursive relation that depends on the previous two terms. Applying the formula for the rest, we find the remaining probabilities to be:

p7 = 76.6%
p8 = 51.1%
p9 = 30.4%
p10 = 15.9%
p11 = 7.06%
p12 = 2.54%

So in other words, starting with a clean piece of rare gear, we have a 2.54% chance of enhacing it to lvl 12, or approximately 1 in 39 items will enhance successfully to lvl 12. If you're really interested, here are the probabilities for higher levels:

p13 = 0.690%
p14 = 0.0651%
p15 = 0.0000114%

I wouldn't trust these numbers though, because I can't imagine how Cygnus obtained these values without sufficient testing. Thanks for reading, and if you're interested, I will do another post on how unlucky you really are when you break 5 items in a row trying to enhance to +8, etc...

sir.. pls continue until +11-12

i understand the way u compute using the formula... but i dont know wat to do next!!

tnx sir.......

[Tomatoman]

Originally Posted by KholdStare88

Necromancing, but I think I have a helpful post regarding enhancement chances. In this post we will try to find the probability of starting at lvl 0 (clean) and obtaining level of your choice, which may be interesting for some of you to read.

Let us define the following variables

p# = probability of going from lvl 0 to lvl #
i# = probability of going from lvl (# - 1) to lvl #
r# = eventual rate of success from lvl (# - 1) to lvl # without decreasing
d# = eventual rate of failure and decreasing from lvl (# - 1) to lvl (# - 2)

When I refer to "raw", that means the values given on Cygnus' table. Now I will explain what these variables mean and how they differ from the raw values listed in the table.

p8: this is what we're trying to find, the probability of achieving level 8 from lvl 0 without breaking
i7: this is assuming if you already achieved level 6, what is the probability you will eventually after unlimited tries get to level 7 without breaking, and it is simply p7/p6 (note this probability accounts for decreasing as many times as you want)
r8: assuming we have unlimited trials, the fail rate while staying at the same level doesn't matter since we will repeat the trial, hence r8 = .4/(.4+.21+.15), using values directly from Cygnus' table
d8: similar to r8, we define d8 = .21/(.4+.21+.15)

We are ready to start our analysis. Without much surprise to anyone, we have:

p1 = 100%
p2 = 100%
p3 = 100%
p4 = 100%
p5 = 100%
p6 = 100%

If there is no chance of breaking, then naturally with unlimited trials you will eventually achieve success. In other words, you can get to lvl 6 without breaking and everybody knows this.
______________________________

Now we consider p7. Since your item cannot decrease while getting to lvl 7, we don't have that to worry about. This means we only have to worry about raw success vs. breaking chance. Hence:



So starting with a clean rare item, the probability you'll get it to lvl 7 is about 3/4 of of time, which means on average you'll break 1 out of 4.

For p8, it becomes more interesting. We will have to break this into parts.

a) First, let's assume no decreasing or breaking. Then the probability of getting from lvl 7 to lvl 8 is simply p7r8, where we multiply the probability of successfully achieving lvl 7 by the eventual rate of (7 -> 8).

b) Second, we consider the case were we decrease once then success, or (7 -> 6 -> 7 -> 8). This probability is d8i7 * p7r8, and it's not hard to understand. The d8 accounts for decreasing to lvl 6, and the i7 accounts for going back to lvl 7. Then we multiply because of "AND" by the probability in part (a), which is (7 -> 8).

c) But that's not all, since it can go (7 -> 6 -> 7 -> 6 -> 7 -> 8), and in fact it can decrease to 6 as many times as it wants. Therefore, we will need to use an infinite series to represent p8, which is the sum of all these discrete scenarios. In other words:



Recall that when you decrease to lvl 6 twice, you square the d8i7 term, but you leave the p7r8 alone since you will decrease twice, get back to lvl 7 twice, but get to lvl 8 only once. And we will add on all the other cases. Now, we can factor out p7r8 from this equation:



And using the formula for convergence of infinite series, we obtain:


______________________________

For p9, it is very similar with one difference. The astute reader will notice that in the previous example, i7 = p7/p6, but p6 is 100%, so in fact i7 = p7. Now that's no longer the case, because now i8 will also account for decreasing from 8 -> 7 -> 6 -> (then all variations with finally) -> 9. So therefore, we have:



And we can now generalize this formula to:



Note that i# depends the p# and p(# - 1), which means we have a recursive relation that depends on the previous two terms. Applying the formula for the rest, we find the remaining probabilities to be:

p7 = 76.6%
p8 = 51.1%
p9 = 30.4%
p10 = 15.9%
p11 = 7.06%
p12 = 2.54%

So in other words, starting with a clean piece of rare gear, we have a 2.54% chance of enhacing it to lvl 12, or approximately 1 in 39 items will enhance successfully to lvl 12. If you're really interested, here are the probabilities for higher levels:

p13 = 0.690%
p14 = 0.0651%
p15 = 0.0000114%

I wouldn't trust these numbers though, because I can't imagine how Cygnus obtained these values without sufficient testing. Thanks for reading, and if you're interested, I will do another post on how unlucky you really are when you break 5 items in a row trying to enhance to +8, etc...

Holy crap just skimming through that hurts my head, how long did it take you to write that?

[Xiaozilu]

Originally Posted by KholdStare88

Necromancing, but I think I have a helpful post regarding enhancement chances. In this post we will try to find the probability of starting at lvl 0 (clean) and obtaining level of your choice, which may be interesting for some of you to read.

Let us define the following variables

p# = probability of going from lvl 0 to lvl #
i# = probability of going from lvl (# - 1) to lvl #
r# = eventual rate of success from lvl (# - 1) to lvl # without decreasing
d# = eventual rate of failure and decreasing from lvl (# - 1) to lvl (# - 2)

When I refer to "raw", that means the values given on Cygnus' table. Now I will explain what these variables mean and how they differ from the raw values listed in the table.

p8: this is what we're trying to find, the probability of achieving level 8 from lvl 0 without breaking
i7: this is assuming if you already achieved level 6, what is the probability you will eventually after unlimited tries get to level 7 without breaking, and it is simply p7/p6 (note this probability accounts for decreasing as many times as you want)
r8: assuming we have unlimited trials, the fail rate while staying at the same level doesn't matter since we will repeat the trial, hence r8 = .4/(.4+.21+.15), using values directly from Cygnus' table
d8: similar to r8, we define d8 = .21/(.4+.21+.15)

We are ready to start our analysis. Without much surprise to anyone, we have:

p1 = 100%
p2 = 100%
p3 = 100%
p4 = 100%
p5 = 100%
p6 = 100%

If there is no chance of breaking, then naturally with unlimited trials you will eventually achieve success. In other words, you can get to lvl 6 without breaking and everybody knows this.
______________________________

Now we consider p7. Since your item cannot decrease while getting to lvl 7, we don't have that to worry about. This means we only have to worry about raw success vs. breaking chance. Hence:



So starting with a clean rare item, the probability you'll get it to lvl 7 is about 3/4 of of time, which means on average you'll break 1 out of 4.

For p8, it becomes more interesting. We will have to break this into parts.

a) First, let's assume no decreasing or breaking. Then the probability of getting from lvl 7 to lvl 8 is simply p7r8, where we multiply the probability of successfully achieving lvl 7 by the eventual rate of (7 -> 8).

b) Second, we consider the case were we decrease once then success, or (7 -> 6 -> 7 -> 8). This probability is d8i7 * p7r8, and it's not hard to understand. The d8 accounts for decreasing to lvl 6, and the i7 accounts for going back to lvl 7. Then we multiply because of "AND" by the probability in part (a), which is (7 -> 8).

c) But that's not all, since it can go (7 -> 6 -> 7 -> 6 -> 7 -> 8), and in fact it can decrease to 6 as many times as it wants. Therefore, we will need to use an infinite series to represent p8, which is the sum of all these discrete scenarios. In other words:



Recall that when you decrease to lvl 6 twice, you square the d8i7 term, but you leave the p7r8 alone since you will decrease twice, get back to lvl 7 twice, but get to lvl 8 only once. And we will add on all the other cases. Now, we can factor out p7r8 from this equation:



And using the formula for convergence of infinite series, we obtain:


______________________________

For p9, it is very similar with one difference. The astute reader will notice that in the previous example, i7 = p7/p6, but p6 is 100%, so in fact i7 = p7. Now that's no longer the case, because now i8 will also account for decreasing from 8 -> 7 -> 6 -> (then all variations with finally) -> 9. So therefore, we have:



And we can now generalize this formula to:



Note that i# depends the p# and p(# - 1), which means we have a recursive relation that depends on the previous two terms. Applying the formula for the rest, we find the remaining probabilities to be:

p7 = 76.6%
p8 = 51.1%
p9 = 30.4%
p10 = 15.9%
p11 = 7.06%
p12 = 2.54%

So in other words, starting with a clean piece of rare gear, we have a 2.54% chance of enhacing it to lvl 12, or approximately 1 in 39 items will enhance successfully to lvl 12. If you're really interested, here are the probabilities for higher levels:

p13 = 0.690%
p14 = 0.0651%
p15 = 0.0000114%

I wouldn't trust these numbers though, because I can't imagine how Cygnus obtained these values without sufficient testing. Thanks for reading, and if you're interested, I will do another post on how unlucky you really are when you break 5 items in a row trying to enhance to +8, etc...

omfg they should give this guy an Olympic medal

[Tajirii]

Sadly I must have the worse luck then, I failed 5 times in a row last night trying to go +4 on a Dark Elf Leggings. It was a sad day. Average onyx took a hit. Now I have to collect more. lol

Also, love the math on this, I would have quoted it, but I think the last 3 people to have done that have significantly lengthened the thread, so why add more. LOL!