The transition matrix of a Markov chain is:

[ .96 .04 ]

[ .27 .73 ]

If it starts in state 2, what is the probability that it will be in state 2 after 2 transitions?

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To return to state 2 after 2 transitions, the chain can either move from

i) 2 - 1 - 2

ii) 2 - 2 - 2

The probability of the move 2 - 1 is the lower left value of the transition matrix (row 2 to column 1), 0.27. The probability of the move 1 - 2 is the upper right value (row 1 to column 2), 0.04. The probability of the move 2 - 2 is the lower right value, 0.73.

Add the possible sequences of transitions i)-ii) together to give the total probability of moving from state 2 back to state 2 in 2 transitions:

i) 0.27 x 0.04 = 0.0108

ii) 0.73 x 0.73 = 0.5329

The total probability is then i) + ii) = 0.5437

This can also be ascertained from the 2-step transition matrix obtained by multiplying the transition matrix by itself (using matrix multiplication). This gives

[0.9324 0.0676]

[0.4563 0.5437]

The probability of going from state 2 to state 2 in two moves is the lower right value 0.5437.

**The probability is 0.544 to 3dp**

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