[JBKProductions]

Let Probability of going from 1 to 1. Let's assume there are 4 states {1, 2, 3, 4}. I just want to know by first step analysis why is(for example) ? I don't understand this at all.

[nuodai]

I take it this is a Markov chain of some variety? (i.e. future depends only on present state, not past states.)

Suppose you start at position 3. Then:

(a) You can go to position 1 with probability , in which case you're absorbed to position 1 with probability
(b) You can go to position 2 with probability , in which case you're absorbed to position 1 with probability
(c) You can stay at position 3 with probability , in which case you're absorbed to position 1 with probability
(d) You can go to position 4 with probability , in which case you're absorbed to position 1 with probability

Combining this information we get

Since it's a Markov chain, the reason why all the probabilities "reset" is because when you know your second position it doesn't matter where you started off.

[JBKProductions]

Thanks!