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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 got different answer. any can help? thanks

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Show (at least one of) your answers to part (a). – Did May 7 '13 at 16:40

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}.

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Why should this yield a Markov chain? – Did May 7 '13 at 16:42
Because the probability of transition to a state is dependent on only the previous state and not on the entire history of the chain. – response May 7 '13 at 16:45
This is not true at the state (Both sit together). The transition to (Male at table 1 and Female at table 2) depends on whether Male and Female are both at table 1 or both at table 2. Hence, unless I am missing a miracle somewhere, the lumped process is not Markov. – Did May 7 '13 at 16:59
Both at table 1 or both at table 2 is included in the state "both sit together". – response May 7 '13 at 17:03
Sure, and so what? Please read my previous comment. – Did May 7 '13 at 17:04

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