An ant walks from vertex to vertex of a triangle in a clockwise direction with probability: $\begin{cases} P = \frac{1}{2} \text{ if it's at vertex 1} \\ P = \frac{3}{4} \text{ if it's at vertex 2} \\ P = 1 \text{ if it's at vertex 3} \end{cases}$
What is the PTM of this Markov chain?
I think it has 3 states (the vertices) but the way I'm understanding the probabilities show that rows don't sum to 1 which is most likely incorrect.
For example, I set the transition probability from state $1$ to state $2$ as $\frac{1}{4}$ because that's the probability it takes the ant to travel there?
$\hspace{7.75cm}$1$\hspace{2mm}$ 2$\hspace{2mm}$ 3
$$\begin{bmatrix} 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} \\ \frac{1}{2} & 0 & 0 \end{bmatrix}$$
Please clarify
You also need to consider the alternative cases. It goes to the clockwise state with probability $p$ and counterclockwise with probability $1-p$.
$\hspace{7cm}$
1$\hspace{9mm}$2$\hspace{9mm}$3$$\begin{bmatrix} 0 & \frac{1}{2} & 1-\frac{1}{2} \\ 1-\frac{3}{4} & 0 & \frac{3}{4} \\ 1 & 1-1 & 0 \end{bmatrix}$$