I have a transition matrix
$$\mathbf P = \begin{bmatrix}0.9 & 0.06 & 0.04 \\ 0.6 & 0.3 & 0.1\\0.7 & 0.25 & 0.05\end{bmatrix}$$
The different states are labeled A, I, O in that order of the columns, so that $$P_{00} = P(A | A) = 0.9,$$ for example. I would like to calculate the probability of single paths in the graph.
Specifically, I'm looking for the probability that either path AAA, IAA or OAA is taken (so a 2-step process under these restrictions).
My intuition first told me to calculate $$ P_{00} * (P_{00} + P_{10} + P_{20})$$ but this yields 1.98, which is larger than 1. So my next instinct tells me to divide this by the sum of all 2-step probabilities from A, I, or O to A, which is $$ P^{2}_{00} + P^{2}_{10} + P^2_{20}. $$ After division, this yields approximately 0.798.
Is this the right process, or am I missing something? I feel like it ought to be simpler. Intuitively I'm looking at probability of a set of specific 2-step paths (O to A to A, I to A to A, or A to A to A) given my transition probabilities, over all such possible 2-step paths (O, I, A) to A.
You want $\mathbf P_{00}(\mathbf P_{00}\ p_0+\mathbf P_{10}\ p_1+\mathbf P_{20}\ p_2)$, where $p_0,p_1,p_2$ are the probabilities for the initial state of the system being $A,I,O$.