Refinement lvl Chance of success Compund probability BE needed BE returned for failure 1.................................100.00%......................100.00%..................0.............................0 2.................................100.00%......................100.00%..................0.............................0 3..................................90.00%........................90.00%..................1.............................0 4..................................80.00%........................72.00%..................2.............................1 5..................................80.00%........................57.60%..................4.............................2 6..................................70.00%........................40.32%..................7.............................3 7..................................50.00%.........................20.16%..................10...........................4 8..................................40.00%..........................8.06%..................15...........................5 9..................................40.00%..........................3.23%..................20...........................6
10.................................30.00%..........................0.97%..................25...........................7
11.................................20.00%..........................0.19%..................35...........................8
12.................................10.00%..........................0.02%..................50...........................9

Adapted from http://phpbb.acclaim.com/9dragons/viewtopic.php?t=85639&postdays=0&postorder=asc&highlight=reece&start=15

This post comes follows from the new patch which has introduced new refinement levels.
(This post is still under construction, and will be further edited.)

Expected Cost of A Refined Weapon Without Hardening

I shall first draw up a general formula for calculating the price associated with a weapon of a particular refinement level:
You go to the refiner with your weapon, the refinement material and pay some gold, and there are 2 possible outcomes:
1. You get a weapon of the next refinement level with chance (c+z).
2. You fail and get some BE returned, if any, with a chance [1-(c+z)].
Where z is the added chance of success from your refinement mastery.

Let W be the price of the unrefined weapon,
and B be the price of BE,
M(n) be the number of BE associated with the material required to refine from n to n+1,
Br(n+1) be the number of BE returned upon failure in refining from n to n+1,
Pr(n) be the expected cost of a weapon of a particular refinement level and
R(n) be the amount of gold you have to pay to the refiner to refine it from n to n+1.

Pr(n) + R(n)+ M(n) = (c+z)Pr(n+1) + [1-(c+z)]Br(n+1)

Rearranging, we derive a recursive formula for calculating the expected cost associated with a weapon of refinement n+1.

Pr(n+1) = 1/(c+z){Pr(n) + R(n) + M(n) - [1-(c+z)]Br(n+1)}

With z being a variable, it is sadly not possible to compute the results in explicit form. Hence, as an addition to the previous post, I shall simply compute the expected cost if refinement mastery is maxed, and that will serve as the lower limit to the expected cost of refinement, while the previous post will serve as the upper limit.

Let z = 0.05

Pr(1) = W + 1100 + 1000 = W +2100
Pr(2) = Pr(1) + 3300 + 3000 = W + 8400
Pr(3) = [1/(0.95)][Pr(2) + 5000 + B] = 1.05W + 14105 + 1.05B
Pr(4) = [1/(0.85)][Pr(3) + 10000 + 2B - (0.15)B] = 1.24W + 28359 + 3.41B
Pr(5) = [1/(0.85)][Pr(4) + 20000 + 4B - 2(0.15)B] = 1.46w + 56893 + 8.37B
Pr(6) = [1/(0.75)][Pr(5) + 35000 + 7B - 3(0.25)B] = 1.94W + 122524 + 19.49B
Pr(7) = [1/(0.55)][Pr(6) + 50000 + 10B - 4(0.45)B] = 3.53W + 313680 + 50.35B
Pr(8) = [1/(0.45)][Pr(7) + 75000 + 15B - 5(0.55)B] = 7.85W + 863734 + 139.12B
Pr(9) = [1/(0.45)][Pr(8) + 100000 + 20B - 6(0.55)B] = 17.44W + 2141631 + 346.26B
Pr(10) = [1/(0.35)][Pr(9) + 125000 + 25B - 7(0.65)B] = 49.83W + 6476088 + 1047.74B
Pr(11) = [1/(0.25)][Pr(10) + 175000 + 35B - 8(0.75)B] = 199.33W + 26604352 + 4306.97B
Pr(12) = [1/(0.15)][Pr(11) + 250000 + 50B - 9(0.85)B] = 1328.89W + 179029013 + 28995.44B

Expected Cost of A Refined Weapon With Hardening

You go to the refiner with your hardened weapon, the refinement material and pay some gold, and there are 2 possible outcomes:
1. You get a weapon of the next refinement level with chance (c+z).
2. You fail with a chance [1-(c+z)]. You get the weapon back and lose 1 hardness point.
Note that no BE is returned upon failure.

Let H be the cost associated with 1 hardness point.

Pr(n) + R(n)+ M(n) = (c+z)Pr(n+1) + [1-(c+z)][Pr(n)-H]

As before, we rearrange the equation to obtain a recursive formula for calculating the expected cost associated with a weapon of refinement n+1, when hardening is involved.

Pr(n+1) = [1/(c+z)][(c+z)Pr(n) + R(n)+ M(n) + [1-(c+z)]H]

As before, we assume that refinement mastery is maximised.