Math, Materia, and How to Succeed at Melding

[Alzelia]

Recently I've seen people complaining that the numbers given for materia melding don't seem right. I believe that the numbers are accurate. I don't have any hard evidence, (I think I'm gonna start taking notes now) but I have always had a fascination with probability and gambling. I have analyzed the materia melding system with my background in statistics and thought I would share my results, and write a guide for materia melding. Don't like math? skip to the blue text.

This started as a response I wrote on one a thread a couple months ago, that I have expanded to this guide.

Materia and Probability

Here's some math for you guys....

With a 20% chance of success, what are the odds that you would fail 5 attempts in a row? 10? 20? 100? 1000?

Let n = the number of attempts at forbidden melding
Let S = The probability of success for a single meld = .2
Let F = The probability of every attempt failing

F(n)=(1-S)^n
F(n)=(1-.2)^n
F(n)=.8^n
F(5)=.8^(5)= .32
F(10)=.8^(10)=.107
F(20)=.8^(20)= .0115
F(100)=.8^(100)= 2.03 x 10^(-10)
F(1000)=.8^(1000)= 1.23 x 10^(-97)

So What does this mean?

There is a 32% that every attempt out of 5 attempts will fail. That is almost a one in three chance. Out of 20 attempts to meld, there is a 1% chance that ever single one of them will fail. You have a better chance of winning the lottery 4 times in a row than you do of failing 1000 attempts in a row though.

But here's the most interesting one to me, if you do 10 double melds in a row, at 20% chance of success each, you will still have a 1 in 10 chance that you could fail every attempt.

Math, Materia, and How to Succeed at Melding

There are three common ways to get a Triple Melded gear
1)Buy it from the wards
2)Get lucky
3)Go for multiple attempts to increase the chances one of them will succeed.

So if you don't want to rely on luck then you can either make one or buy one. Most of the time it would be cheaper just to buy it. The math is a little complicated to determine how much you can expect to spend, but I'll give you the example to give you an idea how much it really costs to triple meld something.

The Actual Cost of Melding

I made this example for this guide. I decided to see how much it would really cost for me to have a triple melded item if luck was not involved. I had a ton of mind materia, so I decided to try for a triple meld of mind iii (12 mind). I used a Woolen Hat NQ for this test. The first materia is always 100%. The second mind iii had a 26% chance of a double meld. Once I was able to get a double meld, I found that I had a 12% chance for the triple meld. (side note: a Mind iv (20 Mind) had a 9.19 % chance for the same meld).

The first thing I have to do is determine the number of double melds I need in order to "expect" to get one item.

1/(26%)=1/(.26)= 3.8 hats

Let me explain what I mean by "expect." If there is a 26% chance for successful double meld, then in the long run we "expect" the actual percentage to eventually approach 26% as the number of attempts approaches infinity. If it does, then on average, it will take 3.8 meld attempts to succeed a double meld. Now I will do the same thing with the percentage for the triple meld.

1/(12%)=1/(.12)= 8.3 double melded hats.
(8.3)(3.8)=31.6 so lets round that up to 32 hats

What this means is that you should expect to have to use:

72 mind iii materia
72 catalysts
32 hats

in order to successfully triple meld a woolen hat with mind iii (12) materia.

HQ, NQ, and melding

I'm have yet to collect actual data for HQ vs. NQ melding. Hopefully I can soon. So let's say there is a flat 5% bonus for HQ gear that is to be double melded, and a 2% bonus for HQ gear that is to be triple melded. I believe this is accurate. Below are the example percentages I will use for this excersize.

HQ
1st:100%
2nd:30%
3rd:10%

NQ
1st:100%
2nd:25%
3rd:8%

Im Skipping writing out the math for right now, but using the method above, we know that you can expect on average to use 33 pieces of HQ gear per successful triple meld. However with the lower percentages for NQ gear you can expect to use 50 NQ gear on average per successful triple meld. NQ gear has a lower chance of melding, which means it takes more gear and materia in order to be able to expect a triple meld. Below is an example of this.

HQ cost:
10 Int IV materia
66 int iii materia
76 catalyst
33 HQ hats

NQ cost
12.5 Int IV materia
100 int iii materia
113 catalyst
50 NQ hats

So if you are using expensive materia, I highly recommend only double/triple melding HQ gear.

Melding Order

For this example I'm using mind iii materia yet again. Mind iii materia can have either 11, 12, 13, or 15 mind. I set up an experiment to test the question, should I meld the better materia first or the worse? I tested every possible combination and noticed that if I tried to add a 11 materia to a hat a 15 materia, my success chance was 24%. If I tried to meld a 15 mind materia with a hat that had 11 materia, my chance of success was 28%. Thus it is better to meld the lower quality materia to an item before a higher quality one. In other words you should add the 11 mind materia first then to double meld the 15 mind materia.

TLDR

What all this boils down to is this very simple fact:
You wanna take a risk, do it yourself.
You wanna be safe, buy it.