a male and a female go to a $2$-table restaurant on the same day. each day the male sits at one or the other of the $2$ tables, starting at the table $1$, with a Markov chain transition matrix: $$\begin{bmatrix}0.3 & 0.7\\ 0.7 & 0.3\end{bmatrix}$$ similarly the female sits at one or the other of the $2$ tables, starting at the table $2$, with a Markov chain transition matrix: $$\begin{bmatrix}0.4 & 0.6\\ 0.6 & 0.4\end{bmatrix}$$ assume that $2$ chains are independent.
a. model this situation with a three-state Markov chain and transition matrix.
b. find the probability that the male sits at table $1$ and the female sits at table $2$ on day $2,3$ and $4$.
c. if $N$ is the number of days that the male and the female sit the same table, then how can we describe the random variable $N$?
I'm new to markov chain and each time I work out part (a), I get a different answer. Can someone help?
Hint: The possible states for the markov chain are: {Both sit together, Male sites at Table $1$ and Female at Table $2$, Male sits at Table $2$ and Female at Table $1$}.