Chapter 8.3, Problem 30E

(a)

To determine

The absorbing stochastic matrix with states $0,5,1,2,3\text{ and 4}$.

(b)

To determine

To calculate: The stable matrix of the $\begin{array}{l}\begin{array}{llllll}0\hfill & 5\hfill & 1\hfill & 2\hfill & 3\hfill & 4\hfill \end{array}\\ \begin{array}{l}0\hfill \\ 5\hfill \\ 1\hfill \\ 2\hfill \\ 3\hfill \\ 4\hfill \end{array}\left[\begin{array}{llllll}1\hfill & 0\hfill & 0.5\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0.5\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0.5\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0.5\hfill & 0\hfill & 0.5\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0.5\hfill & 0\hfill & 0.5\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.5\hfill & 0\hfill \end{array}\right]\end{array}$.

(c)

To determine

The probability that the particle would be begin at location 3 will eventually be absorbed at location 0.

(d)

To determine

The particle begin at location 4 and find the expected number of times that it will be at location $0$.