Figuring Out the Average Cost of Getting a Risky Armor Refinement Level
by Andrew Sun (AKA Sketches)

For those who don’t want to figure out the math of figuring out the cost of upgrading to a certain refinement level (starting from refinement level 0), I decided to post it up for everyone’s benefit.

Refinement level : Average # of Items needed = Average # of Items needed : # of Elunium needed
+5 : 5/3 = 1.67 : 8.33
+6 : 25/6 = 4.17 : 23.33
+7 : 125/12 = 10.42 : 60.83
+8 : 625/12 = 52.08 : 309.17
+9 : 3,125/12 = 260.42 : 1,550.83
+10 : 15,625/6 = 2,064.17 : 15,518.33

Flip the fraction around and you have your chance of success upgrading from +4 to that level.

Here’s the formulas in simple terms:

Variables Defined:
Cost of Single Item = I
Cost of Single Elunium (including refining costs)= E

+5
Cost = 5/3[I + E(5)]

+6
Cost = 25/6[I + E(5.6)]

+7
Cost = 125/12[I + E(5.84)]

+8
Cost = 625/12[I + E(5.936)]

+9
Cost = 3,125/12[I + E(5.9552)]

+10
Cost = 15,625/6[I + E(5.95904)]

Example:
You know those +10 Tidal Boots you’ve always wanted? Well if Tidal Boots are 650,000 zeny and Elunium is 40,000 then the plug in is this:
Cost = 15,625/6[650,000 + 40,000(5.95904)]
so
Cost = 2,313,441,667 zeny
Ouch! Over 2 billion zeny for +10 Tidal Boots. Even if you hunt the boots and elunium yourself and break those instead, you’re still losing money when you break them since you could have sold the boots (opportunity costs).

And that’s pretty much it. On a side note, Level 4 weapons also follow the same formula without a Whitesmith. Since I’m too lazy to post up the formula of level 4 weapons for a job level 70 Whitesmith, I’ll just give the number of weapons needed for a +10 of different levels.

317.46 weapons are needed on average for a job level 70 Whitesmith to make a +10 level 4 weapon in comparison to 2,064.17 by yourself.
88.18 weapons for a +10 level 3 weapon in comparison to 416.67.
31.75 weapons for a +10 level 2 weapon in comparison to 104.17.
9.52 weapons for a +10 level 1 weapon in comparison to 20.83.
But don’t stick this into a Whitesmith’s face. They leveled up to that level so they deserve to charge a lot because of that reason so don’t bug them to make you a level 10 weapon when you have all the materials unless you’re willing to tip them GENEROUSLY for attempting (you should pay them even if the weapon fails to create). Remember that they could have probably made a nice sum of zeny in the time they took to help you out, not to mention the fact that they can refuse to make your weapon that normally sells for an insane price, which is a good reason to refuse to make your weapon and instead let you try by yourself the 1/2064 chance instead of the 1/317 chance with a Whitesmith.

Now for the for those that want an explanation behind how the formula works, get ready to have a 10 minute math class with Sketches:

Now we know the chance of refinement can be found here:
http://ratemyserver.net/index.php?page=misc_table_refine
Individually the chance of refinement to that level (1 being a 100%):
+5 : 0.6
+6 : 0.4
+7 : 0.4
+8 : 0.2
+9 : 0.2
+10 : 0.1

So now the chance of refinement when going straight from +4 to the selected levels is:
+5 : 0.6 = 0.6 or 60%
+6 : 0.6 * 0.4 = 0.24 or 24%
+7 : 0.6 * 0.4 * 0.4 = 0.096 or 9.6%
+8 : 0.6 * 0.4 *0.4 *0.2 = 0.0192 or 1.92%
+9 : 0.6 * 0.4 *0.4 *0.2 *0.2 = 0.00384 or 0.384%
+10 : 0.6 *0.4 *0.4 *0.2 *0.2 *0.1 = 0.000384 or 0.0384%

Now the average number of armor needed to upgrade is found by 1 divided by the chance of refinement:
+5 : 1/0.6 = 1.667
+6 : 1/0.24 = 4.167
+7 : 1/0.096 = 10.417
+8 : 1/0.0192 = 52.083
+9 : 1/0.00384 = 260.417
+10 : 1/0.000384 = 2604.167

Now say the armor of choice was Silk Robes which you valued for 300k and Elunium for 40k (to make the problem simpler I’m including refining costs in the value of Elunium so let’s say you bought the Elunium for 38k each and the refining cost is 2k). For the amount you spend on the armor, it would be the refinement level’s average number of armor needed multiplied by 300k.
Example:
I wanted a +7 Silk Robe so 10.417 * 300,000 = 3,125,100 zeny
or if I wanted to be more exact:
1/0.096 * 300,000 = 3,125,000 zeny spent on Silk Robes.

Now the amount of zeny spent on Elunium is a bit more complicated. The easiest way to explain it would be to go one by one through the refinement list. The minimum number of Elunium required for +5 Silk Robes is 5. So the average cost of Elunium for +5 Silk Robes is the number of armor needed multiplied by the minimum number of Elunium needed which itself is multiplied by the cost of Elunium.
Example:
I wanted a +5 Silk Robe so 1.667 * (5 *40,000) = 333,400 zeny
I put it in parenthesis to make it more clear, but they are not needed here.
or if I wanted to be more exact:
1/0.6 * (5 * 40,000) = 333,333.33 zeny spent on Elunium

For +6 Silk Robes, we know that some of the Silk Robes broke going to +5 on average, so we have to take the number of the average of those that upgraded and multiply that by the number of armor needed for +6 Silk Robes. The number of the average of those that upgraded successfully is the same as the chance of upgrading to that level multiplied by the number of the armor needed for +6 Silk Robes.
Example:
I wanted a +6 Silk Robe so 1/0.24(5 * 40,000) + 1/0.24(0.6 * 40,000) = 933,333.33 zeny spent on Elunium

For +7 Silk Robes, following the same logic, we just have to know we’re taking the success of the success now.
I wanted a +7 Silk Robe so 1/0.096(5 * 40,000) + 1/0.096(0.6 * 40,000) + 1/0.096(0.24 * 40,000) = 2,433,333.33 zeny spent on Elunium

Notice how we can do some factoring and simplify these problems a bit more:
Average Cost of Elunium for +6 Silk Robe
1/0.24(5 * 40,000) + 1/0.24(0.6 * 40,000) = 933,333.33
can instead be
1/0.24[40,000(5+0.6)] = 933,333.33
which can instead be
1/0.24[40,000(5.6)] = 933,333.33

Average Cost of Elunium for +7 Silk Robe
1/0.096(5 * 40,000) + 1/0.096(0.6 * 40,000) + 1/0.096(0.24 * 40,000) = 2,433,333.33
can instead be
1/0.096[40,000(5+0.6+0.24)] = 2,433,333.33
which can instead be
1/0.096[40,000(5.84)] = 2,433,333.33

Which in this same way we can find the cost of +10 armor in Elunium. Now we can put the two equations together for +7 Silk Robe (the armor cost and the elunium cost).
If 1/0.096 * 300,000 is the armor cost and 1/0.096[40,000(5.84)] is the elunium cost:
1/0.096 * 300,000 + 1/0.096[40,000(5.84)] = 5,558,333.33
can instead be
1/0.096[300,000 + 40,000(5.84)]
which is
Average number of armor needed * [value of armor + value of Elunium*(magical number explained in above paragraphs)]

I hope this explains the formulas I provided.

On a side note, this cost formula is why I found my guild mate selling five +7 chain mail[1] for 2,000,000 crazy on his part and good for the buyer. Let’s figure out the cost of making a +7 chain mail[1]. If we value chain mail[1] at 150k and Elunium at 35k, then:
Cost = 125/12[I + E(5.84)]
Cost = 125/12[150,000 + 35,000(5.84)]
Cost = 3,691,666.67 zeny in cost of making.
So yes, he’s selling at a loss.
But hold on one sec!
The elunium itself costs:
Cost of Elunium = 125/12[35,000(5.84)]
Cost of Elunium = 2,129,166.67 zeny
Yes this means the Eluniums it took to upgrade to +7 were more expensive than the price he offered the chain mail[1] for, so he was losing zeny.

Let’s say my guild mate decided to sell the chain mail to the npc and resold all the Elunium he bought for 33k each.
Assuming he had average luck, he had 10.417 chain mail[1] so he made (32,500 * 10.417) 338553.5 zeny from the armor and (125/12[33,000(5.84)]) 2,007,500 zeny from elunium, so 2,346,052.5 zeny he could have gotten instead for one +7 chain mail instead of 2,000,000 zeny.

I sometimes see people selling things at a loss like this, so I’m just pointing this for the people that plan to sell upgraded equipment.

I may have made a typo as this was a lot of numbers to punch in, so please be kind when pointing it out. I hope this was somewhat useful to know. ;-)

Filed under: Other RO Writings, Tips & Tricks