$K= \begin{pmatrix} 0.3 & 0.3 & 0.4\\ 0.2 & 0.7 & 0.1\\ 0.2 & 0.3 & 0.5 \end{pmatrix}$ is the transition matrix of a markov chain, state space $S=\left\{0,1,2\right\}$.
Calculate $P(X_{12}=2, X_{16}=2 \mid X_0=0)$
I have big problem solving this because I think you need program for it..? :(
But what I'm sure about is before you can calculate $P(X_{12}=2, X_{16}=2 \mid X_0 = 0)$, you need to calculate $P(X_{16}=2 \mid X_0=0)$ because I don't see direct way calculating it without knowing this first.
For $P(X_{16}=2 \mid X_0=0)$ I think I don't have a clear / good calculation because I wrote little program in java but it doesn't give precise results.. maybe they don't need to be as precise though. I have
$$P(X_{16}\mid X_0=0) = K^{16}_{0,2}$$ for $K^n$ we get approx this matrix when we start from $16$:
$$\begin{pmatrix} 0.2222 & 0.5 & 0.2778\\ 0.2222 & 0.5 & 0.2778\\ 0.2222 & 0.5 & 0.2778 \end{pmatrix}$$
I have already trouble with this first calculation which we probably need to for the second but I can't even do it it is very brutal.. : /
Any way to do it more easy and precise?
\begin{align} P(X_{16}=2, X_{12}=2|X_0=0) &= P(X_{16}=2|X_{12}=2, X_0=0) P (X_{12}=2|X_0=0) \\ &=P(X_{16}=2|X_{12}=2)P(X_{12}=2|X_0=0)\\ &=P(X_{4}=2|X_0=2)P(X_{12}=2|X_0=0) \end{align}
Compute $P(X_{4}=2|X_0=2)$ and $P(X_{12}=2|X_0=0)$ and then multiply them.