So I have been given following information: I have two people sitting in two rooms. Let they be called R1 and R2. At a discrete steps of time, I pick an adult at random from either of the rooms and let it decide where to sit next- the chosen adult can decide to stay or move with equal probability. Let's consider a Markov chain with states denoting the number of adults sitting in R1. What is the transition probability matrix?
My attempt: So let the states be {0,1,2}
I want to work on each transition probability matrix element separately.
P(00) = 0.5 (which means initially both of the adults were in R2, and the one I chose decided to stay in R2) P(01) = 0.5 (the chosen one from R2 decided to move to R1) P(02) = 0 (both of them cannot go to R1 together)
But I am confused about elements of the second row. Any hint on how to compute that? I am confused because for example: P(10) means initially, the two adults were in separate room and then the chosen adult has to be from R1 and he decided to move to R2, so P(10) = 0.5, but what about P(11) and P(12).