# Markov Chain Conditional Probability

A Markov chain has the transition probability matrix as follows. $$To$$ $$From \begin{matrix} STATES& 0 & 1 & 2 \\ 0 & 0.6 & 0.3 & 0.1 \\ 1 & 0.3 & 0.3 & 0.4\\ 2 & 0.4 & 0.1 & 0.5\\ \end{matrix}$$ Assume that the initial value $X_0$ has the distribution: $P [X_0 = 0] = 0.3 , P [X_0 = 1] = 0.4$, and$P[X_0 = 2] = 0.3$

Find $P[X_0 = 0, X_1 = 2, X_2 = 1]$

I am just starting to learn how this stuff works. From my understanding I am finding $P[0.6, 0.4, 0.1]$ right? I need some help understanding how to move around the map.

The transition probability matrix tells you the probability of $X_n$ to be at state $k$ given that the previous time ($n-1$) you where at state $j$. So the probability you want is: $$P(X_0=0,X_1=2,X_2=1)=0.3\times 0.1\times 0.1$$
Note that $0.3$ is the probability that comes from the initial distribution.
The way of working with the transition matrix is: look at the transition matrix and see if you are in state $1$ for example go to the line that is state $1$ (in this matrix is the second row) and then if you want to go for example to state $0$ then go to the column of $0$ (in this matrix is the first one).
• There is no state $(0,0)$. At time $0$ you want to be at state $0$ so from the initia distribution find the appropriate probability. Now at time $1$ you want to go at state $2$ so go to transition matrix. But that at time $0$ you where at state $0$, this defines the row of the matrix that should look (row 1) and the fact that you want to go to state $2$ defines the column of the matrix (3 column). So you see that the probability is 0.1 Feb 24 '14 at 20:16