Mongoose, the confusion

Okay, so we know what EJ said, but some friends and I weren't really happy with their explanation or their forumla.  So we played around a little on our own.

BEAR:

Since what we want to know is "How many times do I have to do this task with a %p success rate in order to get %r successes, we can use a negative binomial distribution with r=1 since we only care about the first success.  And we know that p=0.0417 for bear

Therefore, we get (1-0.0417)/(0.0417) = 22.9808

So this tells us that on average it takes 23 attacks for mongoose to proc the first time.

So it takes 21.5625 seconds for 23 attacks to occur.  

So since mongoose lasts 15 seconds, the total time for 1 occurance of mongoose is 36.56 which means that mongoose was active 41.03% of the time, which gives an average of 49.23 agility.  

And since once mongoose is over, we start back again at the "state" we started in, the same equation holds for the next proc.

CAT:

For catform, we know that p=0.0167 so plugging in we get (1-0.0167)/(0.0167) = 58.8802

So this tells us that it takes 59 attacks for mongoose to proc the first time, which means it takes 47.104 seconds for 59 attacks to occur.

Since mongoose lasts 15 seconds, the total time for 1 occurance of mongoose is 62.1 which means that mongoose was active 24.15% of the time, which gives an average of 28.9855 agility.

Which means that mongoose is much better for bear, but really sucks for cat.

And I would like to give special thanks to Gurggy, he did the real work and Rusted for keeping on my ass with questions so we got them answered.

Enchants - Mongoose

From Elitist Jerks:

(1-0.75^ppm should be correct if I'm thinking clearly - 15/60 sec uptime is 45/60 downtime on a 1ppm mongoose proc)

You get 1.25 ppm in cat form (15 specials, 1/60 chance to proc each). 
That should mean 1-0.75^1.25 = 30% uptime for 120 agi -> 36 agi avg (+some haste).

As far as I know, you only get one try to proc mongoose per swipe, right? 
You get 2.67 ppm in bear form (40 specials, 2.5/60 chance to proc each).
1-0.75^2.67 = 46% uptime -> 55 agi avg (+some haste)

Me:

Okay, so using a little math and explaining where it came from, we start with the equation

1-(a)^b where 

a is the percent of downtime you have in a minute and 

b is the number of chances to proc in a minute.

And from Wowwiki: Proc per minute

And for catform, since energy regens at 20 energy per two seconds, you can only get 15 special attacks in a minute using all the tools in your toolbox.  So that is where you get 15 specials in 1 minute or 15 in 60 seconds so 15/60 so that is 60+15 = 75 hits per minute. And Procs per hit = Weapon speed * Procs per minute / 60 which is 1.67% for catform. So 75*0.0167 = 1.2525 = 1.25   

So now you plug that into the formula and get 1-(.75)^1.25 = .302 or 30% of the time (18 seconds out of every minute), you will have the mongoose bonus.  So we know that mongoose grants 120 agility, so that is approximately 2 agility per second, so you would get a bonus average of +36 agility plus some haste, which is slightly better than the +35 agility.

Now we can do the same math for bear.  But here you can get in a lot more specials in a minute since you are really only limited by the global cooldown. (I assume you have enough rage)  So since the global cooldown is 1.5 seconds 60/1.5 = 40.  And Procs per hit = Weapon speed * Procs per minute / 60 which is 4.17% for bearform. So you get 40 specials + 24 normals attacks in a minute which is 64 and 64*4.17% = 2.67 and so 1-(.75)^2.67 = 54% uptime. And if it is up 54% of the time, that is 32.4 seconds and so at an average of 2 agility per second, gives 65 agility average.  Can this be right?

In short, mongoose is good for bear, about equal for kitty.