The given system has the following transition matrix

P = 1/3 1/3 1/3 1/2 0 1/2 0 0 1 .

We must calculate the number of steps to absorption starting in state 2. To do this, we need the fundamental matrix. First, S is the 22 matrix

S = 1/3 1/3 1/2 0 .

Thus,

Q	=	(I-S)-1
	=	2/3 -1/3 -1/2 1 -1
	=	2 2/3 1 4/3 .

The number of steps to absorption, assuming the system starts in state 2 is the sum of the entries in row 2: 1 + 4/3 = 7/3.

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