ok, I'm a bit of a math geek, and I am placing my faith in this community, hoping that at least one or two people won't think I'm totally pathetic for spending my spare time this way. Although I'm at work, and have limited ways to amuse myself. I probably should mention, that any of this may be wrong, because I haven't had as much coffee as I normally drink, but I present it nonetheless, for your edification and amusement, (or merciless mockery if I've made mistakes).
Since I play a monk, I decided to figure out how much of a benefit flurry of blows provides against an opponent of a given AC. I'm selfish, and my character get two attacks normally, so I used that as a starting point. First, I tried to figure out the likelihood of getting at least one attack, but discarded that as needlessly complicated, and not really terribly informative. Then, I decided to just figure out the average number of additional hits flurry of blows provides. To get this, we must figure out the total number of ways attacks can succeed (the sum of the ways each attack can succeed times 20^number of attacks-1), divided by the total number of possible dice rolls (20^number of attacks), and subtract that from the same calculations using flurry. I came up with:
f(x)=[(400(21+A-2-x)+(400(21+A-2-x)+400(21+A-5-x))/8000]-[(20(21+A-x)+20(21+A-3-x))/400]
where x is the opponent's AC, and A is your highest attack bonus (obviously, these are only defined on the integers). Luckily, since we're multiplying and dividing by a lot of the same things, (i.e 20^n) it simplifies nicely to:
f(x)=(15+A-x)/20
Basically, this means that the higher your attack bonus and smaller your opponent's AC, the larger benefit Flurry gives. HOWEVER, this function is only valid when the AC is equal to or greater than A+3 and less than or equal to A+14, because otherwise we end up with negative possible hits, which doesn't make sense. Criticals mean that there is no time when a hit is guaranteed or impossible. Thus, (if we actually care) we have to define the function piecewise, for AC values defined relative to your highest attack bonus. (I should also mention all of this doesn't take into account a number of feats like circle kick or improved critical).
SO...
For x=(-inf,A-3], f(x)=.95
For x=[A-2,A-1], f(x)=(16+BAB-x)/20
For x=A, f(x)=(41-2x)/20
For x=[A+1,A+2], f(x)=.85
For x=[A+3,A+14], f(x)=(15+A-x)/20
For x=[A+15,A+17], f(x)=0
For x=[A+18 ], f(x)=-.05
For x=[A+19], f(x)=0
For x=[A+20,inf.), f(x)=.05
In situations, when you will hit all the time, except for critical failure, Flurry provides the most help, giving 19/20 additional hits on average. Interestingly, when you are fighting something with an AC between 15 and 17 or exactly 19 points higher than your attack bonus, Flurry of blows gives no advantage; when you are fighting something with exactly 18 points higher than your attack bonus, you shouldn't use Flurry of Blows (actually, you should probably run away, but . . . .)
If you're a bit higher level, and get three attacks normally, you can use f(x)=(13+A-x)/20, for four attacks (11+A-x)/20, with a restricted domain, and if you aren't lazy like me, you can derive the other values without too much trouble. Basically, for each additional normal attack, flurry of blows is slightly less useful. Also, if I wasn't lazy, I could use summation notation to generalize the solution. Maybe if I'm bored tomorrow. . .
