In the exercises of Chapter 4, the ninth problem says that
In a sequence of independent flips of a fair coin that comes up heads with probability .6, what is the probability that there is a run of three consecutive heads within the first 10 flips?
Here I define the state variable $X_n$ to represent the number of heads that have come up at the $n_{th}$ toss, so the process $\{X_n, n\geq 0\}$ is a Markov Chain with states set $\{0,1,2,3\}$ and its transition probability matrix is
$P = \left[\begin{matrix} 0.4 & 0.6 & 0 & 0 \\ 0.4 & 0 & 0.6 & 0 \\ 0.4 & 0 & 0 & 0.6 \\ 0 & 0 & 0 & 1 \end{matrix} \right]$
So the answer should be $P_{0,3}^{10} = 0.7014$, but the answer given in the book is $0.5078$. So is this answer wrong or did I construct the wrong transition probability matrix?