I have a problem that can be modeled using the binomial distribution.
I want to know the probability that over n trials I will see k successes with a probability of success $P_s$ and a probability of failure $P_f$
This is pretty straightforward stuff:
$$ \binom{n}{k} {P_s}^k (P_f)^{n-k} $$
However there is also a probability that an event will occur within the trials that subtracts from the total number of successes. This event is exclusive of both a success and a failure, and the probability is represented by $P_r$
Put more simply, in each trial there are three possible outcomes:
Success S : counts as 1 success
Failure F : counts as 0 successes
Reduce R : counts as -1 successes
My first approach to this problem would be to generate the table of all possible outcomes (at length $3^k$) and compute the probability of occurrence for each combination that satisfies the requirements, however this approach is obviously suboptimal
How can I adjust this formula in such a way that it accomodates scaling subtractive cases appropriately?
The answer I went with in this case was actually to look at all of the possible outcomes through simulation.
Essentially for every sample of size N, I computed the likelihood of that sample's occurrence, and added it to the end score