For a Markov chain $\lbrace X_n, n\ge0\rbrace$ with transition probabilities $P_{i,j}$, consider the conditional probability that $X_n = m$ given that the chain started at time $0$ in state $i$ and has not entered state $r$ by time $n$, where $r$ is a specified state not equal to either $i$ or $m$. We are interested in whether this conditional probability is equal to the $n$ stage transition probability of a Markov Chain whose state space does not include state $r$ and whose transition probabilities are

$$Q_{i,j} = \frac{P_{i,j}}{1 - P_{i,r}}, i,j \neq r$$

We want to either prove the equality

$$P\lbrace X_n = m \mid X_0 = i, X_k \neq r, k = 1,\dots, n\rbrace = Q_{i,m}^n$$

or provide a counter example.

Initially I thought that the two quantities are equal, but later I was able to find a counterexample, but still is not able to get the intuition behind the two quantities not being equal.

`closer'' to $r$. Here`

closer'' should be understood metaphorically and not necessarily literally in terms of graph distance. $\endgroup$1more comment