This is my first guide, and I'm sure I'll do a terrible job and make it too short. I could probably also use some feedback.
Here is a fusion calculator: http://www.tstr.no/sh/fusion.php
If you put in a common and a flawless, for example, let's say you put in a common and flawless Chaos Armor. This would give you a 30% epic chance. So let's say you have 4 fusion slots and want to do two runs (8 total fuses) and want to know what to expect from it?
The binomial distribution will tell you the probabilities of each number of successes - 1, 2, 3, 4, 5, 6, 7, or 8 epics.
Here is a binomial distribution calculator: http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html
n - the number of trials
p - the probability of success
x - pick a number, like 4, if you want to know the probability of that specific number of successes.
It is easiest to leave x blank and look at the graph. It will show you the probabilities for each number of successes. In this case, we would type .3 in for p and 8 in for n.
The average number of successes is p*n. So for 8 fuses with a 30% success rate, you would get an average of 2.4 epics. You can take this number and multiply it by how many gems you plan to make per epic to see how many gems you would make on average with 8 fuses.
I sell Chaos Armor for 40 gems and get 30, so I would expect to see an average of 72 gems (2.4*30) after two runs of 4 fuses each. Since each run takes about an hour and 15 minutes, in this case, I would expect to see 72 gems every 2.5 hours.
Now, let's say that you were doing some sort of cce fuse (with Chaos Armor, why not). You would normally use a lower-level epic with two common Chaos Armors - but let's pretend you use an epic Chaos Armor as well. You would have a 34.34% chance of an epic, sure, but also a 6.87% legendary chance. Now let's say you want to know the number of fuses you will have to make before your first legendary shows up.
This would use the geometric distribution, which can be found here: http://keisan.casio.com/exec/system/1180573193
Plug .0687 in for the probability of success, and let's say you want to know what the probability of getting a legendary within 16 fuses is - with this particular calculator, you would type in 15 (number of trials - 1, or the number of failures)
In this case, the graph shows you the probability of getting your legendary after exactly 0 failures (1st try), 1 failure (2nd try), etc.
The "lower cumulative p" will tell you the total chance of getting at least one legendary by the time you have finished 16 fuses. It would be around 68%.
The average number of trials it will take you to get your first legendary is 1/p, so in this case, that would be 1/.0687, or around 14.6 fuses.
Note: I think most people use lower-level items for the flawless in the first example and especially for the epic in the second example.