I have a Finite State Machine represented in following form:
[ 0, 1, 0, 0, 0, 1,]
[ 4, 0, 0, 2, 3, 0,]
[ 0, 0, 0, 0, 0, 0,]
[ 0, 0, 0, 0, 0, 0,]
[ 0, 0, 0, 0, 0, 0 ]
Where each row represents a state (first row is s0, second one is s1, etc) and the values represent the probability of transitioning to the next state. For instance, s0 has 50% chance of going to s1 and 50% chance of going to s5.
How can I calculate probability of going from s0 to s3? If these states did not have a loopback (you can go from s0 to s1 and then back to s0 according to this matrix) answer would be trivial, but I am bit confused about how this loop or feedback changes the probabilities.
It looks as if $s_2,s_3,s_4$ all absorb and that $s_2$ is inaccessible.
Let $p$ be the probability you want. Then
$$p = \frac12\times\left(\frac49\times p +\frac29 \times 1 + \frac39 \times 0\right)+\frac12 \times 0$$ which is easy to solve for $p$