So I have this graph:
where the initial state is $s_0$ and the only two terminal states are $s_2$ and $s_4$. The probability of the next state is given by the fraction on the arrows.
My task is to find the probability of reaching a terminal state (either $s_2$ or $s_4$) starting from $s_0$.
Empirically, I have found that $p(s_2) = 2/3$ and $p(s_4) = 1/3$.
I understand how to calculate the probability if there are no cycles (by simply tracing and multiplying the probabilities) or just one cycle (by tracing and using converging geometric series) but I don't understand how to calculate the probability if the cycles are intertwined (e.g. $s_1$ and $s_0$ create a cycle which is intertwined with the cycle $s_0, s_1, s_3$)
Any ideas would be very much appreciated!