My question is: Starting in state 1 what is the probability on ending up in state 3 resp. state 4?
The transition matrix T= \begin{bmatrix} 0& 0.8& 0&0.2\\ 0&0.2&0.6&0.2\\ 0&0&0&0\\ 0&0&0&0 \end{bmatrix} Both state 3 and 4 are absorbing states, and once you have left state 1 you can not get back there. At first I calculated it to be 0.6 chance to end up in state 3 and 0.4 for state 4. I got this using
$\pi_0=0.8\pi_1+0.2\pi_3$,
$ \pi_1=0.2\pi_1+0.6\pi_2+0.2\pi_3$,
$\pi_2=\pi_2$,
$\pi_3=\pi_3$
$\pi_0+\pi_1+\pi_2+\pi_3=1$
Setting $\pi_0=0$ as is not possible to end up there.
However, recalculating it I get something looking very wrong $47/20\pi_2+29/20\pi_3=1$
I also would like to know how many steps on average does it take to get to state 3?