Exercise: n$-$step calculation

$0\mathrm{.}0/2\mathrm{.}0$ points $($graded$)$

Consider the following transition probability graph:

This figure depicts a Markov chain with three possible states. The possible states are S$\_1,$ S$\_2,$ and S$\_3,$ which are depicted as a row of circles on the middle of the diagram and placed from left to right in this order. At the upper part of the diagram, there are self$-$loops within S$\_1,$ S$\_2,$ and S$\_3,$ which are circular arrows with both the tail and the head at the same state. In this figure, transitions are depicted by arrows from a circle $($representing the source$)$ to a circle $($representing the destination$).$ The source and the destination may be the same, and transition arrows may be labelled with a number or a variable indicating the probability that if we start at the source, we will be at the given destination at the next step. The probabilities for each of the self$-$loops are $0\mathrm{.}6$ for S$\_1,0\mathrm{.}7$ for S$\_2,$ and $0\mathrm{.}4$ for S$\_3.$ Between the circles representing the states are transitions between S$\_1$ and S$\_2,$ as well as transitions between S$\_2$ and S$\_3.$ The transition probability from S$\_1$ to S$\_2$ is $0\mathrm{.}1,$ and that from S$\_2$ to S$\_1$ is $0\mathrm{.}2.$ The transition probability from S$\_2$ to S$\_3$ is $0\mathrm{.}1,$ and that from S$\_3$ to S$\_2$ is $0\mathrm{.}3.$ Below the row where the states are presented, the transitions between S$\_1$ and S$\_3$ are shown. The transition probe ability from S$\_1$ to S$\_3$ and from S$\_3$ to S$\_1$ are both $0\mathrm{.}3.$

Calculate the three$-$step transition probability by using the recursion formula $.$