I'm having trouble with Markov chains

Question

Below is a photo of the problem I'm having trouble with.

Below is my work up to this point, however 0.38 is showing up as incorrect. I'm honestly not sure where I'm going wrong here. Any help would be greatly appreciated!

Answer 1

The phrasing of the question is a bit tricky. They ask "what proportion of the state 2 population will be in state 2 after two steps?". This is equivalent to asking "given that we start in state 2, what is the probability of landing in state 2 after 2 steps?" In other words, you should look for the second entry of $$ P^2 \pmatrix{0\\1} = \pmatrix{0.54\\0.46}. $$ So, your answer should be $0.46$.

Answer 2

The question seems to be asking "what proportion of the state 2 population will be in state 2". This is $P(X_2=2\mid X_0=2)$, which you can read off from the matrix $P^2$ -- it is the $(2,2)$ element, which by your calculation is $0.46$.