Markov chain problem

[Dawazeezee]

Hey I came across this question. The topic is matrices and it says:

Let P =

0.5 0.3
0.5 0.7

be the transition matrix for a Markov chain with 2 states.

Let x0=
0.5
0.5

be the initial state vector for the population

What proportion of the state 2 population will be in state 2 after two steps? The answer is 64%.

I'm a bit confused here. What exactly is this question asking for? What do they want me to calculate? Thanks

 

[royhaas]

Look at \(\displaystyle P^2\).

 

[Dawazeezee]

Yes, P^2 is

2/5 9/25
3/5 16/25

and when multiplied by

0.5
0.5

I got

.38
.62

which gives state 2 as 62%, but the answer is 64%. What's wrong?