Markov Model transition probability

Question

Hy, i have a little doubt about a Markov model problem. The problem requests to find a transition probability matrix for a situation with two statistically independent person that can be in 4 different state $S_1, S_2, S_3, S_4$. The transaction probability for one person is:

state 1 goes directly back to state 1 with probability 0.4 and to state 2 with probability 0.6. State 2 goes directly to states 1 or 4 with probabilities 0.2 and 0.8, respectively. State 3 goes directly to states 1, 3, or 4 with probabilities 0.1, 0.7, and 0.2, respectively. State 4 goes directly to states 1, 3, or 4 with probabilities 0.3, 0.2, and 0.5, respectively.

I've constructed the model with the notation $(X,Y,Z,W)$ finding 16 different states. For example state $(AB,0,0,0)$ indicate the two person in the state $S_1$. $(A,B,0,0)$ indicate the person $A$ in $S_1$ and person $B$ in $S_2$ and so on. Enumerated the states I have to find the transition probability but, what is the transition probability from state $(AB,0,0,0)$ to $(A,B,0,0)$? How can I find it having only that for one person?

Bye

Answer 1

If each individual visits states from a state space $S$, the couple of individuals visits states from $S\times S$. By independence, the transition probability from $(x,x')$ in $S\times S$ to $(y,y')$ in $S\times S$ is $Q((x,x'),(y,y'))=q(x,y)q(y,y')$, where $q$ denotes the transition matrix of each individual considered separately.