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 0.6 for S_1, 0.7 for S_2, and 0.4 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 0.1, and that from S_2 to S_1 is 0.2. The transition probability from S_2 to S_3 is 0.1, and that from S_3 to S_2 is 0.3. 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 0.3.
Calculate the three-step transition probability
r11(3) by using the recursion formula rij(n)=3∑k=1 rik(n−1)pkj.
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