Probability that Markov chain state is not touched for at least n steps

Question

Consider a Markov chain with two states (1,2):

$$P={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}$$

I want the probability that after a state 1, state 1 is not touched for at least N consecutive steps (i.e. only state 2 is touched for at least N consecutive steps).

I think that the answer should be

$$ P_{12}(1-\sum_{k=1}^{N-1}P_{22}) $$

But I'm not sure if this is correct.

---

EDIT thanks a lot for the answers, I made a mistake in the question and I apologize, the answer that I meant was

$$ P_{12}(1-\sum_{k=1}^{N-1}P_{22}^k) $$

Is this still incorrect?

Answer 1

You formula is not correct.

The answer should be $P_{12} P_{22}^{N-1}$, the first step is move from state 1 to state 2, then the chain has to stay in state 2 for the next $N-1$ steps.

Answer 2

This can be expressed as: 1. go from state 1 to 2, and stay there for at least N-1 units of time. ie we have: $p_{12}p_{22}^{N-1}$