NOTE: All graphs on this page have both a raster (image) version and a link to the SVG version. It's recommended to look at the SVG version, as it scales, and, in some browsers, you can get additional information when you hover over stuff. Firefox stubbornly refuses to accept the SVG version, but IE and Chrome work fine.
[Image: http://wiki.mabinogiworld.com/images/7/72/Special_Upgrade_Screen.png]
[SIZE="1"]^ Yeah, right?[/SIZE]
[SIZE="4"]Success Rates[/SIZE]
Special upgrade success rates are not as straightforward to calculate as repair or enchant rates. This is because the probability changes dynamically based on your current upgrade level. Also, because it's a +/-1 system, the success rates change the more upgrades you do. As if that wasn't enough, certain conditions clamp the probability to 0 or 1, like a weapon being destroyed upgrading from 5 to 6.
Because of this, I took a statistical approach to calculating success rates. I wrote a small program in Python to model a user upgrading a weapon, according to the in game process and success rates. This includes early termination by either achieving the desired level, or having your item poof. This essentially gave me a way to try upgrades without totally blowing my bank to hell and back.
By using this function to simulate a large number of "upgrades" (I used 1,000,000), we can get a pretty good idea of how the probabilities work out. I used this method to calculate the success rate from 0 to each subsequent level, for stone counts ranging from 1 to 200, which is what this first graph shows:
http://xcelled.rydian.net/ug/ug.svg
[Image: http://i.imgur.com/EmOsDVY.png]
Here's some ways to read this graph:
- With 20 stones, you have a 45% chance of successfully obtaining level 5.
- In order to be fairly assured of getting level 5 (95% success rate), you need about 80 stones.
- If you're exceedingly unlucky, you can expect to need about 160 stones for level 5.
And some things to note:
- The probability curves are logarithmic, meaning you need to use exponentially more stones to increase success rates. Notice how getting the last 5% success for level 5 requires the same number of stones as the first 95%? This is why special upgrades suck horribly if you fail any.
- The small, flat areas in the lines are caused by the +/- 1 mechanic, more on this later.
- The chance of going from 0 to 5 in 5 stones is almost 5%. The math (which is simple for this case because the number of stones is equal to the desired level, meaning we have to get all successes) backs this up: 100% * 50% * 50% * 45% * 45% = 5.06%
- Due to item destruction, rank 6 caps off at 45%.
- It's always more likely to have an item destroyed than to get to rank 6 (obviously).
[SIZE="4"]+/- 1 System[/SIZE]
In addition to chewing up stones, the +/- 1 system has another interesting effect. Some upgrade levels are not obtainable for a given stone count and target level. For example, here are the possible results from using 10 stones to get from 0 to level 5:
[CODE]
Level 0: 25.43%
Level 1: 0.00%
Level 2: 42.51%
Level 3: 0.00%
Level 4: 13.99%
Level 5: 18.04%
Level 6: 0.00%
Destroyed: 0.00%[/CODE]
This works out because there's no combinations of successes/failures with 10 stones that will leave you at 1 or 3. For math nerds, this is because 10 modulo 3 = 1, meaning when you arrive at level 1 or 3, you will always have at least one other stone to use up. The reason 5 is not 0% is that the algorithm stops upon obtaining level 5.
[SIZE="4"]Level distribution[/SIZE]
Finally, have you ever asked, "I have 12 upgrade stones, I wonder what the chances are that I'll obtain level X are?" If so, you'll like the next section. I have compiled a sampling of each level and stone count, similar to above, but this graph uses a stacked bar chart to show you the proportions of each level relative to the others at that stone count. Bigger bars mean you're more likely to get that level.
This may sound complicated, but here's a simple example: If you look at the chart for target level 5, you'll see that for 5 stones, we have:
- 0% chance of level 0
- 63.21% chance of level 1
- 0% chance of level 2
- 31.74% chance of level 3
- 0% chance of level 4
- 5.04% chance of level 5
- 0% chance of level 6
- 0% chance of destruction
You can also look at chart 5 for 10 stones to see the results mentioned in the +/-1 section.
I highly recommend using Chrome to look at the SVGs for these graphs, as that will enable you to see the actual percentage by hovering over the bars.
[SPOILER="Six graphs"]
http://xcelled.rydian.net/ug/ug_distro_1.svg
[Image: http://i.imgur.com/j0QuWr0.png]
http://xcelled.rydian.net/ug/ug_distro_2.svg
[Image: http://i.imgur.com/itVPH9K.png]
http://xcelled.rydian.net/ug/ug_distro_3.svg
[Image: http://i.imgur.com/A6yUQ4t.png]
http://xcelled.rydian.net/ug/ug_distro_4.svg
[Image: http://i.imgur.com/hpxpjDA.png]
http://xcelled.rydian.net/ug/ug_distro_5.svg
[Image: http://i.imgur.com/HuRTk6J.png]
http://xcelled.rydian.net/ug/ug_distro_6.svg
[Image: http://i.imgur.com/ukuvdnR.png]
[/SPOILER]