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For a Markov chain $\{X_n, n\ge0\}$ 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

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

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.

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@joriki : To some folks this seems intuitively correct. Please visit the link – RIchard Williams Dec 26 '12 at 14:37
Sorry, my comment was wrong; I've removed it. You should have linked to the cross-post at MO right away instead of wasting my time with a question that had already been answered elsewhere. And you still haven't linked to this thread at MO. – joriki Dec 26 '12 at 15:21

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