i have a 3 state (0,1,2) markov chain with transition matrix p=

1/2, 1/3, 1/6

0, 1/3, 2/3

1/2, 0, 1/2

and that p(x0= 0) = p(x0= 1)= 1/2

and i need to fond E[X3]

so i find p^3 by matrix multiplication to be:

13/36, 11/54,47/108

4/9, 4/27, 11/27

5/12, 2/9, 13/36

now the solution says the answer is 1/2(47/108)+1/2(11/54)+1/4(13/36)

but i have no idea how this is obtained! please help!