Confused about definition of absorption probability

Question

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?

Answer 1

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.

Answer 2

NOT AN ANSWER; continuation of my Comment. (See Answer just now by @GrahamKemp.)

Computation in R. (Mathematical software may do the job more elegantly.)

 P = matrix(c(1/5, 1/5, 1/5,  0,  2/5,
                0,  1/3,  0,  2/3,  0,
                0,   0,  1/2,  0,  1/2,
                0,  3/5,  0,  2/5,  0,
                0,   0,  1/2,  0,  1/2), byrow=T, nrow=5)

 P2 = P %*% P;  P4 = P2 %*% P2;  P8 = P4 %*% P4;  P8
 ##          [,1]      [,2]      [,3]      [,4]      [,5]
 ## [1,] 2.56e-06 0.1184156 0.3749978 0.1315838 0.3750003
 ## [2,] 0.00e+00 0.4736977 0.0000000 0.5263023 0.0000000
 ## [3,] 0.00e+00 0.0000000 0.5000000 0.0000000 0.5000000
 ## [4,] 0.00e+00 0.4736721 0.0000000 0.5263279 0.0000000
 ## [5,] 0.00e+00 0.0000000 0.5000000 0.0000000 0.5000000

 rowSums(P8)  # reality check
 ##  1 1 1 1 1

 P2 = P %*% P;  P4 = P2 %*% P2;  P8 = P4 %*% P4

 round(P8, 4)
 ##      [,1]   [,2]  [,3]   [,4]  [,5]
 ## [1,]    0 0.1184 0.375 0.1316 0.375
 ## [2,]    0 0.4737 0.000 0.5263 0.000
 ## [3,]    0 0.0000 0.500 0.0000 0.500
 ## [4,]    0 0.4737 0.000 0.5263 0.000
 ## [5,]    0 0.0000 0.500 0.0000 0.500

Already for $\text{P}^8,$ we have 'almost-absorption' probabilities $.1184 + .1316 = .25$ and $.375 + .375 = .75,$ correct to four places. Absorption by the 8th transition is all but certain. (Not rounding to 4 places, the "0" in at upper-left is $2.56 x 10^{-06}.$)

This approach may help you understand what is going on, even if it is not a mathematically acceptable answer in your course. (Notice, for example, that exact 0's throughout $\text{P}^8$ show lack of communication between the two closed classes.)

Answer 3

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 $1/5$.