# Is it possible to determine the probability of NOT reaching a state in a FSA ?

Suppose I have a Finite State Automata with various states and probabilities of state transitions. Does mathematics exist to determine the probability of NOT reaching a state in the FSA given some n number of state changes?

First note that the chance of not being in the state exactly at transition $n$ is the complement of the chance of being in the state at transition $n$.
The chance of never encountering the state at some point during the first $n$ transitions is almost as simple to find. Modify the FSA by redirecting all outgoing transitions from that state back to itself so it becomes a terminal (absorbing) state. The chance of being in that state at transition $n$ in the modified FSA is the chance of encountering that state at some point during the first $n$ transitions in the original FSA. The complement of this chance is the desired probability.
• You compute the $n^\text{th}$ power of the transition matrix and inspect the relevant entry. For small $n$ this is elementary, as illustrated at stats.stackexchange.com/questions/45920. A standard, more sophisticated method diagonalizes the transition matrix: see stats.stackexchange.com/questions/87385.