Question:
An experimental animal can stay in room-A until 1 minute,and it can stay in room-B until 2 minutes. There exist deadly gases in room-C. One room among these three rooms is being randomly chosen per a minute. And The animal goes to this chosen room.
How to write down transition matrix?
Answer
State space is
S ={ 1minute in room-A, 1minute in room-B, 2minutes in room-B, room-C}={0, 1, 2, 3} (respevtively)
Transition matrix has been written like that;
$$\mathcal P = \begin{bmatrix} 1/3 & 1/3 & 0 & 1/3 \\ 0 & 0 & 1 & 0 \\ 1/3 & 1/3 & 0 & 1/3 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
But, I cannot understand how to write down this transition matrix?
Why $P_{00}^{(1)}=1/3$ or $P_{12}^{(1)}=1$ or ...? Please how to write these probability values in the matrix? Explain it with reasons. thank you.