I am struggling to get started on this question.
I think I am confused at what the transition matrix is suppose to represent.
So I know the matrix is going to have this form: $$ \begin{vmatrix} P_{00}&P_{01}&&...&P_{0M}\\ P_{10}&P_{12}&&...&P_{1M}\\ .&.&&.&.\\ .&.&&.&.\\ .&.&&.&.\\ P_{M1}&P_{M2}&&...&P_{MM}\\ \end{vmatrix} $$
Is this sufficient for the transition matrix for general M?
To clear up my confusion, what does $P_{01}$ mean? I think this means that A starts with 0 molecules, and something happened to make it have 1 molecule? (The molecule must have been removed from urn B and put into A, but how do we know urn B has molecules?) How do I find the probability of $P_{01}$? Also what information allows us to populate the matrix with values?
This is a model by T. Ehrenfest for the movement of molecules in which M molecules are distributed among two urns.
I quote:
If we let $X_n$ denote the number of molecules in the first urn immediately after the $n$th exchange, then $\{X_0, X_1,…\}$ is a Markov chain with transition probabilities
$$P_{i, j+1} = \frac{M-i}{M} \:\:\:\:\:\: 0 \leq i \leq M$$
$$P_{i, j-1} = \frac{i}{M} \:\:\:\:\:\: 0 \leq i \leq M$$
$$P_{i, j} = 0 \:\:\:\:\:\: \text{if}\:\: |j-i| > 1$$