How to calculate the probability of the outcome of this convoluted dice rolling mechanic? This question asks about the probability of a success in a Role Playing Game.  However, the question, and its answers do not cover some of the complexities of the dice mechanic. In particular, it does not cover botches (one possible outcome) at all.
A player has a dice pool, based on some mechanic in the game irrelevant to this question.  A dice pool is a variable number of dice a player may roll.  There are rules about how many dice the player gets to roll, but that is irrelevant to this question.  It can be any number of dice from 1 (a single die) to about 15.  I am calling this P.
The dice have 10 sides labeled 1 through 10 inclusive (Called a 'd10' in our domain terminology)
When rolling dice, there is a target number, or difficulty number.  How this number is generated is outside the scope of this question, but the number can be between 3 and 9 inclusive.  The rules around this are explained below.  I am calling this T.
When all the dice are rolled, there are some rules to determine the outcome:


*

*Any die equal to or greater than T is counted as a success

*Any die equal to 1 subtracts from successes


Such that...


*

*If, after subtraction (if applicable), there are no die left greater than or equal to T, then the result is a failure.

*If, after subtraction (if applicable), there is at least one die left greater than or equal to T, then the result is a success.

*If no die rolled are greater than or equal to T, and at least one die is 1, then it is a botch


For a given P pool and T target, how do you calculate the probability of a success, failure, or botch in this system?
 A: I'll have to tackle this in stages as time permits. I expect someone will give a complete (and probably simpler) approach before I finish. 
First, let's look at botches.
I'm going to ignore some of your notation and call the number of dice $n$. 

If no die rolled are greater than or equal to T, and at least one die is 1, then it is a botch

First consider $P(\text{no dice }\geq T) = (\frac{T-1}{10})^n$
Now consider $P(\text{no } 1| \text{no dice }\geq T) = (\frac{T-2}{T-1})^n$
So $P(\text{botch}) = [1- (\frac{T-2}{T-1})^n]\cdot (\frac{T-1}{10})^n$
$\hspace{2cm} = \frac{(T-1)^n-(T-2)^n}{10^n}$
(assuming I didn't make any errors)

Second, the distribution of the number of individual-die successes after subtraction can be tackled by the method at this post. However, you seem to be after $P(\text{at least one success in total})$ (i.e. the overall roll succeeds) which I think may be amenable to relatively more simple-minded approaches (though they may well involve more work in the end). I'll look at that next edit. 
