Question 5.4.1: A group insurance plan allows three different options for pa...
A group insurance plan allows three different options for participants, plan A, B, or C. Suppose that the percentages of the total number of participants enrolled in each plan are 25 percent, 30 percent, and 45 percent, respectively. Also, from past experience assume that participants change plans as shown in the table.
| C | B | A | |
| 0.2 | 0.25 | 0.75 | A |
| 0.4 | 0.45 | 0.15 | B |
| 0.4 | 0.3 | 0.1 | C |
a. Find the percent of participants enrolled in each plan after 5 years.
b. Find the steady-state vector for the system.
Let T be the matrix given by
T = \begin{bmatrix} 0.75& 0.25& 0.2\\0.15& 0.45& 0.4\\0.1 &0.3 &0.4 \end{bmatrix}
a. The number of participants enrolled in each plan after 5 years is approximated by the vector
T^{5} v = \begin{bmatrix} 0.49776 &0.46048 &0.45608\\ 0.28464& 0.30432 &0.30664\\ 0.21760 &0.23520& 0.23728 \end{bmatrix} \begin{bmatrix} 0.25\\ 0.30\\ 0.45 \end{bmatrix} = \begin{bmatrix} 0.47\\ 0.30\\0.22 \end{bmatrix}so approximately 47 percent will be enrolled in plan A, 30 percent in plan B, and 22 percent in plan C.
b. The steady-state vector for the system is the probability eigenvector corresponding to the eigenvalue λ = 1, that is,
S = \begin{bmatrix} 0.48 \\ 0.30 \\ 0.22 \end{bmatrix}