Confused about definition of absorption probability My confusion can probably most easily be explained with an example. Consider the following one step transition matrix :
$$
  P=\matrix{%
    & 0 & 1  & 2  & 3   & 4 \\
    0 & 1/5  & 1/5 & 1/5 & 0 & 2/5 \\
    1 & 0 & 1/3  & 0 & 2/3 & 0 \\
    2 & 0 & 0 & 1/2 & 0  & 1/2 \\
    3 & 0 & 3/5 & 0 & 2/5 & 0 \\
    4 & 0 & 0 & 1/2 & 0 & 1/2
  }$$(I annotated each column and row with the respective corresponding state)
Clearly there are two closed classes $\{2,4\}, \{1,3\}$, and one open class consisting of the single transient state $0$. Now what I don't understand is, say I was looking for the probability of absorption into $\{1,3\}$ from $0$. 
Is this simply $1/5$, or do you ignore the probability of remaining in the transient state ($1/5$), and recalculate accordingly to have it be $1/4$? 
Or more generally, does "probability of absorption into class $X$" ignore the probability of remaining in the transient state, or do you take the probability directly?
 A: 
Is this simply 1/5 , or do you ignore the probability of remaining in the transient state (1/5 ), and recalculate accordingly to have it be 1/4 ? 

If you are in state 0 at the start of a step, you can immediately either transition into $\{1,3\}$ (via state $1$, with probability $\tfrac 1 5$), remain in state $0$ ( with probability $\tfrac 1 5$), transition into either entry into $\{2,4\}$ (with probability $\tfrac 3 5$).  If you remain, you repeat this in the next step.  If you enter $\{2,4\}$ you will never reach $\{1,3\}$.  
So the probability of eventually entering into the closed class $\{1,3\}$, is $$\overbrace{\qquad\qquad\qquad\tfrac 1 5 +}^{\text{immediately enter }\{1,3\}} \overbrace{\tfrac 1 5(\tfrac 1 5+ \tfrac 1 5(\tfrac 1 5 + \tfrac 1 5(\ldots)))\qquad\qquad\qquad}^{\text{immediately remain in } 0\text{, but eventually enter }\{1,3\}} = \frac{\tfrac 1 5}{\tfrac 1 5 + \tfrac 3 5} = \tfrac  1 4$$
tl;dr Yes, you have to recalibrate, if you want to find the probability of eventually entering a state rather than immediately.
A: This seems more like a question about terminology than the underlying mathematics (which you seem to understand just fine).  Absent any other context, I would understand "the probability of absorption into $\{1, 3\}$" to refer to the probability of absorption into that pair of states eventually (and therefore the probability is $1/4$); in order to interpret it differently, I would expect it to be phrased as something like "the probability of immediate absorption into $\{1, 3\}$," and then the probability is $1/5$.
