Chapter 9, Problem 26RE

A red urn contains $2$ red marbles, $1$ blue marble, and $1$ green marble. A blue urn contains $1$ red marble, $3$ blue marbles, and $1$ green marble. A green urn contains $6$ red marbles, $3$ blue marbles, and $1$ green marble. A marble is selected from an urn, the color is noted, and the marble is returned to the urn from which it was drawn. The next marble is drawn from the urn whose color is the same as the marble just drawn. This is a Markov process with three states: draw from the red urn, draw from the blue urn, or draw from the green urn.

(A) Draw a transition diagram for this process.

(B) Write the transition matrix $P$.

(C) Determine whether this chain is regular, absorbing, or neither.

(D) Find the limiting matrix $\bar{P}$, if it exists, and describe the long-run behavior of this process.