= Holding probabilities cancel from hitting-probability equations
{title2=$h_i=\sum_{j\ne i}\frac{P_{ij}}{1-P_{ii}}h_j\quad(P_{ii}<1)$}
Outside a target set, <first-step analysis> for a <Markov chain> gives $h_i=P_{ii}h_i+\sum_{j\ne i}P_{ij}h_j$. Moving the holding term and dividing gives the displayed average over the next distinct state. Thus inserting or removing self-loops preserves eventual hitting probabilities, provided the state is not made absorbing and the relative probabilities of its exits are unchanged. Boundary values are one on the target and zero in closed classes unable to reach it. The waiting time changes, but the sequence of distinct visited states does not.
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